JEE Advanced Coordinate Geometry Cheat Sheet

This comprehensive cheat sheet is designed to be your ultimate companion for mastering Coordinate Geometry for JEE Advanced. It condenses all essential formulas, concepts, and problem-solving strategies into an easily digestible format, ensuring you're well-equipped to tackle even the most challenging questions.

JEE Advanced Coordinate Geometry Cheat Sheet

Introduction to Coordinate Geometry

Coordinate Geometry, also known as Analytical Geometry, is a fundamental branch of mathematics that establishes a correspondence between geometric figures and algebraic equations. It allows us to describe points, lines, curves, and surfaces using numerical coordinates and to study their properties using algebraic methods. For JEE Advanced, this topic is crucial as it forms the bedrock for understanding many advanced concepts in calculus and 3D geometry. A strong grasp of Coordinate Geometry is indispensable for success in the entrance examination.

Exam Weightage Analysis (JEE Advanced)

Coordinate Geometry consistently carries significant weightage in JEE Advanced. Historically, questions from this section, including 2D and 3D geometry, contribute anywhere from 15% to 25% of the total marks in the Mathematics paper. This makes it one of the most important topics to master. While separate questions are common, integration with other topics like calculus (finding areas, tangents, normals) is also prevalent. Expect approximately 2-4 questions directly from 2D Coordinate Geometry in each paper.

I. Fundamental Concepts & Basic Formulas

1. Distance Formula

The distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is given by: $$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

2. Section Formula

a) Internal Division

If a point $R(x, y)$ divides the line segment joining $P(x_1, y_1)$ and $Q(x_2, y_2)$ internally in the ratio $m:n$, then: $$x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n}$$

b) External Division

If a point $R(x, y)$ divides the line segment joining $P(x_1, y_1)$ and $Q(x_2, y_2)$ externally in the ratio $m:n$, then: $$x = \frac{mx_2 - nx_1}{m-n}, \quad y = \frac{my_2 - ny_1}{m-n}$$

c) Midpoint Formula

The midpoint of the line segment joining $P(x_1, y_1)$ and $Q(x_2, y_2)$ is: $$x = \frac{x_1 + x_2}{2}, \quad y = \frac{y_1 + y_2}{2}$$

3. Area of a Triangle

The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is: $$\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$ Alternatively, using the determinant method: $$\text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{vmatrix} \right|$$ For collinear points, the area of the triangle is 0.

4. Centroid, Incenter, Circumcenter, Orthocenter

Suppose the vertices of a triangle are $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$.

a) Centroid (G)

The centroid is the intersection of the medians. It divides each median in the ratio $2:1$. $$G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$$

b) Incenter (I)

The incenter is the intersection of the angle bisectors. Let $a, b, c$ be the side lengths $BC, CA, AB$ respectively. $$I = \left(\frac{ax_1 + bx_2 + cx_3}{a+b+c}, \frac{ay_1 + by_2 + cy_3}{a+b+c}\right)$$

c) Circumcenter (O)

The circumcenter is the intersection of the perpendicular bisectors of the sides. It is equidistant from all three vertices. Its coordinates $(x,y)$ can be found by solving: $$(x-x_1)^2 + (y-y_1)^2 = (x-x_2)^2 + (y-y_2)^2$$ $$(x-x_2)^2 + (y-y_2)^2 = (x-x_3)^2 + (y-y_3)^2$$

d) Orthocenter (H)

The orthocenter is the intersection of the altitudes. Euler Line: The centroid (G), circumcenter (O), and orthocenter (H) are collinear, and G divides OH in the ratio $1:2$. $$G = \frac{2O + H}{3} \implies H = 3G - 2O$$ This relation is often used to find H if G and O are known.

Summary Table of Triangle Centers

II. The Straight Line

1. Slope of a Line (Gradient)

a) Given two points $P(x_1, y_1)$ and $Q(x_2, y_2)$:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

b) Given the angle $\theta$ a line makes with the positive x-axis:

$$m = \tan \theta$$

c) Given the equation of a line $Ax + By + C = 0$:

$$m = -\frac{A}{B}$$

2. Conditions for Parallel and Perpendicular Lines

a) Parallel Lines

Two non-vertical lines with slopes $m_1$ and $m_2$ are parallel if: $$m_1 = m_2$$

b) Perpendicular Lines

Two non-vertical lines with slopes $m_1$ and $m_2$ are perpendicular if: $$m_1 m_2 = -1 \quad (\text{Note: horizontal line } y=c \text{ and vertical line } x=k \text{ are perpendicular})$$

3. Equations of a Line in Different Forms

a) Slope-Intercept Form

$y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.

b) Point-Slope Form

$y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is its slope.

c) Two-Point Form

$\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}$, passing through $(x_1, y_1)$ and $(x_2, y_2)$. Alternatively: $(y - y_1)(x_2 - x_1) = (x - x_1)(y_2 - y_1)$.

d) Intercept Form

$\frac{x}{a} + \frac{y}{b} = 1$, where $a$ is the x-intercept and $b$ is the y-intercept.

e) Normal Form (Perpendicular Form)

$x \cos \alpha + y \sin \alpha = p$, where $p$ is the length of the perpendicular from the origin to the line and $\alpha$ is the angle this perpendicular makes with the positive x-axis.

f) General Form

$Ax + By + C = 0$, where A, B are not both zero.

4. Angle Between Two Lines

If $m_1$ and $m_2$ are the slopes of two lines, the acute angle $\theta$ between them is given by: $$\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$$

5. Distance Formulas (Lines)

a) Distance of a Point From a Line

The perpendicular distance of a point $P(x_1, y_1)$ from the line $Ax + By + C = 0$ is: $$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$$

b) Distance Between Two Parallel Lines

The distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is: $$d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$$

6. Position of a Point Relative to a Line

A point $P(x_1, y_1)$ lies on one side or the other of the line $Ax + By + C = 0$. The expression $Ax_1 + By_1 + C$ determines the side. If $Ax_1 + By_1 + C$ and $Ax_2 + By_2 + C$ have the same sign, points $(x_1, y_1)$ and $(x_2, y_2)$ lie on the same side of the line. If they have opposite signs, they lie on opposite sides. The origin $(0,0)$ provides a convenient reference point using $C$.

7. Family of Lines

The equation of a line passing through the intersection of two lines $L_1: A_1x + B_1y + C_1 = 0$ and $L_2: A_2x + B_2y + C_2 = 0$ is given by: $$L_1 + \lambda L_2 = 0$$ $$(A_1x + B_1y + C_1) + \lambda(A_2x + B_2y + C_2) = 0$$ where $\lambda$ is a parameter.

8. Reflection of a Point

The reflection of a point $P(x_1, y_1)$ across a line $Ax + By + C = 0$ is $P'(x', y')$. The line segment $PP'$ is perpendicular to the given line, and the midpoint of $PP'$ lies on the line. $$\frac{x' - x_1}{A} = \frac{y' - y_1}{B} = -2 \frac{Ax_1 + By_1 + C}{A^2 + B^2}$$

9. Bisectors of Angles Between Two Lines

For two lines $A_1x + B_1y + C_1 = 0$ and $A_2x + B_2y + C_2 = 0$, the equations of the angle bisectors are: $$\frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}}$$ To identify the bisector of the angle containing the origin, first make $C_1 > 0$ and $C_2 > 0$ (if not, multiply the equation by -1). Then the '+' sign gives the bisector of the angle containing the origin. To identify the acute/obtuse angle bisector: Calculate $A_1A_2 + B_1B_2$. If $A_1A_2 + B_1B_2 > 0$: The '+' sign gives the obtuse angle bisector. The '-' sign gives the acute angle bisector. If $A_1A_2 + B_1B_2 < 0$: The '+' sign gives the acute angle bisector. The '-' sign gives the obtuse angle bisector.

III. Pair of Straight Lines

1. Homogeneous Equation of Second Degree

The equation $ax^2 + 2hxy + by^2 = 0$ represents a pair of straight lines passing through the origin.

The slopes $m_1, m_2$ of these lines are given by $m_1 + m_2 = -\frac{2h}{b}$ and $m_1 m_2 = \frac{a}{b}$. The angle $\theta$ between these lines is given by: $$\tan \theta = \frac{2\sqrt{h^2 - ab}}{|a+b|}$$ Conditions:

• If $h^2 - ab > 0$: lines are real and distinct.
• If $h^2 - ab = 0$: lines are real and coincident.
• If $h^2 - ab < 0$: lines are imaginary (pass through real origin).
• If $a+b = 0$: lines are perpendicular to each other.
• If $h^2 - ab > 0$ and $h=0, a=0$ or $b=0$, this gives degenerate cases (e.g. $xy=0$ gives $x=0, y=0$).

2. General Equation of Second Degree

The equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of straight lines if: $$\begin{vmatrix} a & h & g \ h & b & f \ g & f & c \end{vmatrix} = 0$$ i.e., $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$. If this condition holds, the intersection point of the two lines $(x_0, y_0)$ can be found by solving $\frac{\partial F}{\partial x} = 0$ and $\frac{\partial F}{\partial y} = 0$: $ax + hy + g = 0$ $hx + by + f = 0$ The angle $\theta$ between these lines is the same as for the homogeneous equation: $$\tan \theta = \frac{2\sqrt{h^2 - ab}}{|a+b|}$$

3. Homogenization

To find the equation of lines joining the origin to the intersection points of a line $L: lx + my + n = 0$ and a curve $C: F(x, y) = 0$ (e.g., a circle $Ax^2 + By^2 + Gx + Fy + C = 0$), we homogenize $C$ using $L$. From $L$, we have $1 = -\frac{lx+my}{n}$. Substitute this into $C$ to make it a homogeneous equation of degree two: $Ax^2 + By^2 + Gx\left(-\frac{lx+my}{n}\right) + Fy\left(-\frac{lx+my}{n}\right) + C\left(-\frac{lx+my}{n}\right)^2 = 0$ This will give the equation of a pair of straight lines passing through the origin and the points of intersection.

IV. The Circle

1. Standard Equation

The equation of a circle with center $(h, k)$ and radius $r$ is: $$(x - h)^2 + (y - k)^2 = r^2$$

2. General Equation

The general equation of a circle is: $$x^2 + y^2 + 2gx + 2fy + c = 0$$ Here, center is $(-g, -f)$ and radius is $r = \sqrt{g^2 + f^2 - c}$. Conditions for a real circle: $g^2 + f^2 - c > 0$.

• If $g^2 + f^2 - c = 0$: Point circle (radius zero).
• If $g^2 + f^2 - c < 0$: Imaginary circle.

3. Equation of Circle in Special Cases

a) Circle passing through origin: $x^2 + y^2 + 2gx + 2fy = 0$ (since $c=0$).

b) Circle with center on x-axis: $x^2 + y^2 + 2gx + c = 0$ ($f=0$).

c) Circle with center on y-axis: $x^2 + y^2 + 2fy + c = 0$ ($g=0$).

d) Circle touching x-axis: radius $|k|$. $(x-h)^2 + (y-k)^2 = k^2$.

e) Circle touching y-axis: radius $|h|$. $(x-h)^2 + (y-k)^2 = h^2$.

f) Circle touching both axes: radius $|r|$. $(x \pm r)^2 + (y \pm r)^2 = r^2$. (Four possible circles).

g) Equation of a circle with $P(x_1, y_1)$ and $Q(x_2, y_2)$ as diameter endpoints:

$$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$$

4. Parametric Equation

For $(x - h)^2 + (y - k)^2 = r^2$: $$x = h + r \cos \theta, \quad y = k + r \sin \theta$$ where $\theta$ is the parameter ($0 \le \theta < 2\pi$).

5. Position of a Point Relative to a Circle

For a circle $S \equiv x^2 + y^2 + 2gx + 2fy + c = 0$ and a point $P(x_1, y_1)$:

• If $S_1 \equiv x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c > 0$: Point P lies outside the circle.
• If $S_1 \equiv x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c = 0$: Point P lies on the circle.
• If $S_1 \equiv x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c < 0$: Point P lies inside the circle.

6. Length of Tangent

The length of the tangent from an external point $P(x_1, y_1)$ to the circle $S=0$ is $\sqrt{S_1}$. $$L = \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}$$

7. Tangent to a Circle

a) Equation of Tangent at a Point $(x_1, y_1)$ on the Circle $S=0$:

Replace $x^2 \to x x_1$, $y^2 \to y y_1$, $x \to \frac{x+x_1}{2}$, $y \to \frac{y+y_1}{2}$, $xy \to \frac{xy_1+yx_1}{2}$. For $x^2 + y^2 + 2gx + 2fy + c = 0$, the tangent at $(x_1, y_1)$ is: $$x x_1 + y y_1 + g(x + x_1) + f(y + y_1) + c = 0$$

b) Condition for a Line $y = mx + c'$ to be Tangent to Circle $x^2 + y^2 = R^2$:

$$c'^2 = R^2(1 + m^2)$$ The point of tangency is $\left(\mp \frac{mR^2}{c'}, \pm \frac{R^2}{c'}\right)$.

c) Condition for a Line $y = mx + c'$ to be Tangent to Circle $x^2 + y^2 + 2gx + 2fy + C = 0$:

The distance from the center $(-g, -f)$ to the line $mx - y + c' = 0$ must be equal to the radius $r = \sqrt{g^2+f^2-C}$. $$\frac{|m(-g) - (-f) + c'|}{\sqrt{m^2 + (-1)^2}} = \sqrt{g^2 + f^2 - C}$$

8. Normal to a Circle

The normal to a circle at any point always passes through the center of the circle. Equation of normal at $(x_1, y_1)$ on the circle $S=0$: For $x^2 + y^2 + 2gx + 2fy + c = 0$, the center is $(-g, -f)$. The slope of the radius connecting $(-g, -f)$ and $(x_1, y_1)$ is $m_{rad} = \frac{y_1 - (-f)}{x_1 - (-g)} = \frac{y_1+f}{x_1+g}$. The equation of normal is $y - y_1 = \left(\frac{y_1+f}{x_1+g}\right)(x - x_1)$.

9. Chord of Contact

The equation of the chord of contact of tangents drawn from an external point $P(x_1, y_1)$ to the circle $S=0$ is $S_1=0$. $$x x_1 + y y_1 + g(x + x_1) + f(y + y_1) + c = 0$$ This is the same as the tangent equation, but $(x_1, y_1)$ is an external point.

10. Director Circle

The locus of the point of intersection of two perpendicular tangents to a circle is called the Director Circle. For $x^2 + y^2 = R^2$, the director circle is $x^2 + y^2 = 2R^2$. For $x^2 + y^2 + 2gx + 2fy + c = 0$, the director circle is $(x+g)^2 + (y+f)^2 = 2(g^2+f^2-c)$.

11. Radical Axis

The radical axis of two circles $S_1 = 0$ and $S_2 = 0$ is the locus of points from which tangents to both circles have equal length. Equation of radical axis is $S_1 - S_2 = 0$. For $S_1 \equiv x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $S_2 \equiv x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$: $$2(g_1 - g_2)x + 2(f_1 - f_2)y + (c_1 - c_2) = 0$$ Properties of Radical Axis:

• It is a straight line.
• It is perpendicular to the line joining the centers of the two circles.
• If circles intersect, radical axis is their common chord.
• If circles touch, radical axis is their common tangent.
• If circles are concentric, radical axis is undefined (or at infinity).

12. Radical Center

For three circles $S_1=0$, $S_2=0$, $S_3=0$, the radical axes $S_1-S_2=0$, $S_2-S_3=0$, and $S_3-S_1=0$ are concurrent at a point called the radical center.

13. Family of Circles

a) Passing through the intersection of a circle $S=0$ and a line $L=0$:

$S + \lambda L = 0$ $$x^2 + y^2 + 2gx + 2fy + c + \lambda(Ax + By + C) = 0$$

b) Passing through the intersection of two circles $S_1=0$ and $S_2=0$:

$S_1 + \lambda S_2 = 0$ $$ (x^2 + y^2 + 2g_1x + 2f_1y + c_1) + \lambda(x^2 + y^2 + 2g_2x + 2f_2y + c_2) = 0 \quad (\text{where } \lambda \ne -1)$$ If $\lambda = -1$, it represents the radical axis.

V. Conic Sections (Parabola, Ellipse, Hyperbola)

General Definition of Conic Sections

A conic section is the locus of a point that moves in a plane such that its distance from a fixed point (focus, $S$) bears a constant ratio (eccentricity, $e$) to its distance from a fixed straight line (directrix, $l$). $$\frac{PS}{PM} = e$$

• If $e = 1$: Parabola
• If $0 < e < 1$: Ellipse
• If $e > 1$: Hyperbola
• If $e = 0$: Circle (directrix is at infinity)

1. The Parabola

a) Standard Equation

$y^2 = 4ax$

• Vertex: $(0,0)$
• Focus: $(a,0)$
• Directrix: $x = -a$
• Axis: $y=0$ (x-axis)
• Latus Rectum Length: $4a$ (endpoints are $(a, \pm 2a)$)

Other forms:

• $y^2 = -4ax$: Focus $(-a,0)$, Directrix $x=a$
• $x^2 = 4ay$: Focus $(0,a)$, Directrix $y=-a$
• $x^2 = -4ay$: Focus $(0,-a)$, Directrix $y=a$

b) Parametric Equation

For $y^2 = 4ax$: $x = at^2, y = 2at$. The point is $P(at^2, 2at)$.

c) Tangent to a Parabola

• At a point $(x_1, y_1)$ on $y^2 = 4ax$: $y y_1 = 2a(x + x_1)$
• In terms of slope $m$ for $y^2 = 4ax$: $y = mx + \frac{a}{m}$ (tangency point $(\frac{a}{m^2}, \frac{2a}{m})$)
• In terms of parameter $t$ for $y^2 = 4ax$: $ty = x + at^2$

d) Normal to a Parabola

• At a point $(x_1, y_1)$ on $y^2 = 4ax$: $y - y_1 = -\frac{y_1}{2a}(x - x_1)$
• In terms of slope $m$ for $y^2 = 4ax$: $y = mx - 2am - am^3$
• In terms of parameter $t$ for $y^2 = 4ax$: $y + tx = 2at + at^3$

e) Chord of Contact

From an external point $(x_1, y_1)$ to $y^2 = 4ax$: $y y_1 = 2a(x + x_1)$.

f) Director Circle

Locus of point of intersection of perpendicular tangents to a parabola is its directrix. For $y^2 = 4ax$, it is $x = -a$.

2. The Ellipse

a) Standard Equation

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b)$

• Center: $(0,0)$
• Vertices: $(\pm a, 0)$
• Co-vertices: $(0, \pm b)$
• Foci: $(\pm ae, 0)$
• Directrices: $x = \pm \frac{a}{e}$
• Eccentricity: $e = \sqrt{1 - \frac{b^2}{a^2}}$ or $b^2 = a^2(1-e^2)$.
• Latus Rectum Length: $\frac{2b^2}{a}$
• Major Axis Length: $2a$
• Minor Axis Length: $2b$

If $b > a$ (major axis along y-axis): $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$

• Vertices: $(0, \pm a)$
• Foci: $(0, \pm ae)$
• Directrices: $y = \pm \frac{a}{e}$
• Eccentricity: $e = \sqrt{1 - \frac{b^2}{a^2}}$ (here $a$ represents the semi-major axis, $b$ semi-minor)

b) Parametric Equation

For $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: $x = a \cos \phi, y = b \sin \phi$. The point is $P(a \cos \phi, b \sin \phi)$. $\phi$ is the eccentric angle.

c) Tangent to an Ellipse

• At a point $(x_1, y_1)$ on $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: $\frac{x x_1}{a^2} + \frac{y y_1}{b^2} = 1$
• In terms of slope $m$: $y = mx \pm \sqrt{a^2 m^2 + b^2}$
• In terms of parameter $\phi$: $\frac{x \cos \phi}{a} + \frac{y \sin \phi}{b} = 1$

d) Normal to an Ellipse

• At a point $(x_1, y_1)$ on $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: $\frac{a^2 x}{x_1} - \frac{b^2 y}{y_1} = a^2 - b^2$
• In terms of parameter $\phi$: $ax \sec \phi - by \csc \phi = a^2 - b^2$

e) Chord of Contact

From an external point $(x_1, y_1)$ to $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: $\frac{x x_1}{a^2} + \frac{y y_1}{b^2} = 1$

f) Director Circle

For $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the director circle is $x^2 + y^2 = a^2 + b^2$.

3. The Hyperbola

a) Standard Equation

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

• Center: $(0,0)$
• Vertices: $(\pm a, 0)$
• Foci: $(\pm ae, 0)$
• Directrices: $x = \pm \frac{a}{e}$
• Eccentricity: $e = \sqrt{1 + \frac{b^2}{a^2}}$ or $b^2 = a^2(e^2-1)$.
• Latus Rectum Length: $\frac{2b^2}{a}$
• Transverse Axis Length: $2a$ (along x-axis)
• Conjugate Axis Length: $2b$ (along y-axis)
• Asymptotes: $y = \pm \frac{b}{a} x$

Conjugate Hyperbola: $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$

• Foci: $(0, \pm be)$
• Directrices: $y = \pm \frac{b}{e}$
• Asymptotes remain $y = \pm \frac{b}{a} x$

b) Parametric Equation

For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $x = a \sec \phi, y = b \tan \phi$. The point is $P(a \sec \phi, b \tan \phi)$.

c) Tangent to a Hyperbola

• At a point $(x_1, y_1)$ on $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $\frac{x x_1}{a^2} - \frac{y y_1}{b^2} = 1$
• In terms of slope $m$: $y = mx \pm \sqrt{a^2 m^2 - b^2}$
• In terms of parameter $\phi$: $\frac{x \sec \phi}{a} - \frac{y \tan \phi}{b} = 1$

d) Normal to a Hyperbola

• At a point $(x_1, y_1)$ on $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $\frac{a^2 x}{x_1} + \frac{b^2 y}{y_1} = a^2 + b^2$
• In terms of parameter $\phi$: $ax \cos \phi + by \cot \phi = a^2 + b^2$

e) Chord of Contact

From an external point $(x_1, y_1)$ to $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $\frac{x x_1}{a^2} - \frac{y y_1}{b^2} = 1$

f) Director Circle

For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the director circle is $x^2 + y^2 = a^2 - b^2$.

• If $a^2 > b^2$, it is a real circle.
• If $a^2 = b^2$ (rectangular hyperbola), it degenerates to the point $(0,0)$.
• If $a^2 < b^2$, it is an imaginary circle.

4. Rectangular Hyperbola

A hyperbola where the length of the transverse axis is equal to the length of the conjugate axis, i.e., $a=b$.

• Equation: $x^2 - y^2 = a^2$
• Eccentricity: $e = \sqrt{1+1} = \sqrt{2}$
• Asymptotes: $y = \pm x$ (perpendicular to each other)
• Equation in referred to asymptotes as axes: $xy = c^2$, where $c^2 = a^2/2$.
  • Parametric form: $(ct, c/t)$

5. Tangency Conditions Summary for Conics

VI. Worked Examples

Example 1: Straight Line - Family of Lines

Problem: Find the equation of the line passing through the intersection of the lines $2x + y = 3$ and $5x + y = 6$ and also passing through the point $(1, 2)$.

Solution: The equation of a family of lines passing through the intersection of $L_1: 2x + y - 3 = 0$ and $L_2: 5x + y - 6 = 0$ is given by $L_1 + \lambda L_2 = 0$. So, $(2x + y - 3) + \lambda(5x + y - 6) = 0$. Since this line also passes through the point $(1, 2)$, we substitute $x=1$ and $y=2$ into the equation: $(2(1) + 2 - 3) + \lambda(5(1) + 2 - 6) = 0$ $(2 + 2 - 3) + \lambda(5 + 2 - 6) = 0$ $1 + \lambda(1) = 0$ $\lambda = -1$ Substitute $\lambda = -1$ back into the family of lines equation: $(2x + y - 3) - 1(5x + y - 6) = 0$ $2x + y - 3 - 5x - y + 6 = 0$ $-3x + 3 = 0$ $x = 1$ This is the equation of the required line.

Example 2: Circle - Radical Axis

Problem: Find the equation of the radical axis of the circles $x^2 + y^2 - 4x - 6y + 1 = 0$ and $x^2 + y^2 - 8x - 2y + 13 = 0$.

Solution: Let the first circle be $S_1 \equiv x^2 + y^2 - 4x - 6y + 1 = 0$. Let the second circle be $S_2 \equiv x^2 + y^2 - 8x - 2y + 13 = 0$. The equation of the radical axis is given by $S_1 - S_2 = 0$. $(x^2 + y^2 - 4x - 6y + 1) - (x^2 + y^2 - 8x - 2y + 13) = 0$ $x^2 + y^2 - 4x - 6y + 1 - x^2 - y^2 + 8x + 2y - 13 = 0$ Combine like terms: $(x^2 - x^2) + (y^2 - y^2) + (-4x + 8x) + (-6y + 2y) + (1 - 13) = 0$ $4x - 4y - 12 = 0$ Divide by 4: $x - y - 3 = 0$ This is the equation of the radical axis.

Example 3: Parabola - Normal

Problem: Find the locus of the intersection point of two perpendicular normals to the parabola $y^2 = 4ax$.

Solution: The equation of a normal to $y^2 = 4ax$ with slope $m$ is $y = mx - 2am - am^3$. Let $m_1$ and $m_2$ be the slopes of two perpendicular normals. So $m_1 m_2 = -1$. Let the intersection point be $(h,k)$. Then $P(h,k)$ satisfies the normal equation for both $m_1$ and $m_2$: $k = m_1 h - 2am_1 - am_1^3 \quad (\text{Eq. 1})$ And $k = m_2 h - 2am_2 - am_2^3 \quad (\text{Eq. 2})$ Since $m_2 = -1/m_1$, substitute this into Eq. 2: $k = (-1/m_1)h - 2a(-1/m_1) - a(-1/m_1)^3$ $k = -h/m_1 + 2a/m_1 + a/m_1^3$ Multiply by $m_1^3$: $k m_1^3 = -h m_1^2 + 2a m_1^2 + a$ $a m_1^3 + (h-2a)m_1^2 - k m_1 - a = 0 \quad (\text{Eq. 3})$ From Eq. 1: $am_1^3 + (2a-h)m_1 + k = 0 \quad (\text{Eq. 4})$ Comparing Eq. 3 and Eq. 4 is tricky. Let's start with Eq. 1: $am_1^3 + (2a-h)m_1 + k = 0$ This is a cubic equation in $m$. For any point $(h,k)$, there are three normals (real or complex). Let the slopes be $m_1, m_2, m_3$. We are given that two normals are perpendicular, so $m_1 m_2 = -1$. From Viete's relations for $am^3 + 0 \cdot m^2 + (2a-h)m + k = 0$: $m_1 + m_2 + m_3 = 0$ $m_1 m_2 + m_2 m_3 + m_3 m_1 = \frac{2a-h}{a}$ $m_1 m_2 m_3 = -\frac{k}{a}$

Using $m_1 m_2 = -1$: From $m_1 + m_2 + m_3 = 0$, we have $m_3 = -(m_1 + m_2)$. From $m_1 m_2 m_3 = -k/a$: $(-1) m_3 = -k/a$ $m_3 = k/a$.

Substitute $m_3 = k/a$ into the original normal equation $y = mx - 2am - am^3$. Since the point $(h,k)$ is the intersection point, it must satisfy this for $m_3$: $k = (k/a)h - 2a(k/a) - a(k/a)^3$ $k = \frac{kh}{a} - 2k - a\frac{k^3}{a^3}$ Since $k$ cannot be zero (if $k=0$, then $m_3=0$, which means $y=0$ is a normal, implying $y=0$ is a tangent to $y^2=4ax$ making it $x=0$, but $y=0$ is the axis, not a normal to the parabola unless at the vertex specifically), we can divide by $k$: $1 = \frac{h}{a} - 2 - \frac{k^2}{a^2}$ $a^2 = ah - 2a^2 - k^2$ $k^2 = ah - 3a^2$ Replacing $(h,k)$ with $(x,y)$ to get the locus: $y^2 = ax - 3a^2$ So, the locus is $y^2 = a(x - 3a)$. This is a parabola translated $3a$ units to the right from the standard parabola.

Example 4: Ellipse - Equation using Locus

Problem: An ellipse has its foci at $(\pm 2, 0)$ and its length of the major axis is 6. Find its equation.

Solution: The foci are $(\pm ae, 0)$. Comparing with $(\pm 2, 0)$, we have $ae = 2$. The length of the major axis is $2a$. Given $2a = 6$, so $a = 3$. Now we have $a=3$ and $ae=2$. Substitute $a=3$ into $ae=2$: $3e=2 \implies e = 2/3$. For an ellipse, $b^2 = a^2(1 - e^2)$. Substitute $a=3$ and $e=2/3$: $b^2 = (3)^2 \left(1 - \left(\frac{2}{3}\right)^2\right)$ $b^2 = 9 \left(1 - \frac{4}{9}\right)$ $b^2 = 9 \left(\frac{9 - 4}{9}\right)$ $b^2 = 9 \left(\frac{5}{9}\right)$ $b^2 = 5$ Since the foci are on the x-axis, the major axis is along the x-axis. The standard equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Substitute $a^2 = 3^2 = 9$ and $b^2 = 5$: $\frac{x^2}{9} + \frac{y^2}{5} = 1$ This is the equation of the ellipse.

Example 5: Hyperbola - Asymptotes

Problem: Find the product of the perpendicular distances from any point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ to its asymptotes.

Solution: The equations of the asymptotes for the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ are: $L_1: \frac{x}{a} - \frac{y}{b} = 0 \implies bx - ay = 0$ $L_2: \frac{x}{a} + \frac{y}{b} = 0 \implies bx + ay = 0$ Let $P(x_1, y_1)$ be any point on the hyperbola. So, $\frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} = 1$. This can be rewritten as $\frac{b^2x_1^2 - a^2y_1^2}{a^2b^2} = 1 \implies b^2x_1^2 - a^2y_1^2 = a^2b^2$.

The perpendicular distance from $P(x_1, y_1)$ to $L_1$ is $d_1 = \frac{|bx_1 - ay_1|}{\sqrt{b^2 + (-a)^2}} = \frac{|bx_1 - ay_1|}{\sqrt{a^2 + b^2}}$. The perpendicular distance from $P(x_1, y_1)$ to $L_2$ is $d_2 = \frac{|bx_1 + ay_1|}{\sqrt{b^2 + a^2}} = \frac{|bx_1 + ay_1|}{\sqrt{a^2 + b^2}}$.

The product of these distances is $d_1 d_2 = \frac{|(bx_1 - ay_1)(bx_1 + ay_1)|}{\sqrt{a^2 + b^2} \sqrt{a^2 + b^2}}$. $d_1 d_2 = \frac{|b^2x_1^2 - a^2y_1^2|}{a^2 + b^2}$. Since $P(x_1, y_1)$ lies on the hyperbola, we know that $b^2x_1^2 - a^2y_1^2 = a^2b^2$. So, $d_1 d_2 = \frac{|a^2b^2|}{a^2 + b^2}$. Since $a^2$ and $b^2$ are positive, this simplifies to: $d_1 d_2 = \frac{a^2b^2}{a^2 + b^2}$.

This is a constant product, independent of the point $P(x_1, y_1)$ on the hyperbola.

Example 6: Conics - Intersection and Homogenization

Problem: Find the condition for the two lines given by $ax^2 + 2hxy + by^2 = 0$ to be conjugate diameters of the circle $x^2 + y^2 = r^2$.

Solution: A pair of lines $ax^2 + 2hxy + by^2 = 0$ passes through the origin. These are of the form $y=m_1x$ and $y=m_2x$. Recall that $m_1+m_2 = -2h/b$ and $m_1m_2 = a/b$. For two lines to be conjugate diameters of a circle, they must be perpendicular. If $m_1$ and $m_2$ are the slopes of two lines, for them to be perpendicular, $m_1m_2 = -1$. So, from $m_1m_2 = a/b$, we must have $a/b = -1$. This implies $a = -b$, or $a+b=0$. Thus, the condition for the two lines $ax^2 + 2hxy + by^2 = 0$ to be conjugate diameters of a circle (or perpendicular diameters, since for a circle, all diameters are conjugate if they are perpendicular) is $a+b=0$.

Note that for an ellipse $\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$, two diameters $y=m_1x$ and $y=m_2x$ are conjugate if $m_1m_2 = -\frac{B^2}{A^2}$. The question specifies "circle", for which $A=B=r$. So $m_1m_2=-1$.

VII. Common Mistakes to Avoid

1. Sign Errors: Especially in section formula, distance of point from line, and normal form of a line. Double-check all signs.
2. Transposition Errors: Moving terms across the equals sign in equations often leads to sign errors.
3. Mixing up Formulas: Each conic has distinct formulas for tangent, normal, director circle, etc. Don't use ellipse formulas for hyperbola or vice-versa.
4. Careless Calculation: Simple arithmetic mistakes can ruin an otherwise perfect solution. Practice mental math and double-check calculations.
5. Forgetting Conditions: E.g., $g^2 + f^2 - c > 0$ for a real circle, $h^2 - ab > 0$ for real distinct lines.
6. Incorrectly Applying Homogenization: Remember to form a linear equation from the secant line and use it to make the curve equation homogeneous.
7. Confusing Perpendicular and Parallel Slopes: $m_1 = m_2$ for parallel, $m_1 m_2 = -1$ for perpendicular.
8. Centroid vs. Orthocenter vs. Incenter: Clearly understand the definitions and formulas for each.
9. Normal Form $x \cos \alpha + y \sin \alpha = p$: Ensure $p$ is always positive (distance) and choose $\alpha$ correctly ($0 \le \alpha < 2\pi$).
10. Radical Axis/Center: Remember what $S_1-S_2=0$ represents conceptually (locus of equal tangent lengths).
11. Parametric forms: Understand when and how to use parametric forms for quick calculations and derivation of tangent/normal equations.
12. Forgetting Special Cases: E.g., vertical lines have undefined slope, horizontal lines have slope 0. These cannot be represented by $y=mx+c$.

VIII. Memory Tricks & Mnemonics

• Tangent Condition for Circle $x^2+y^2=R^2$: $y=mx \pm R\sqrt{1+m^2}$ -> Think of it as "radius times 'root-one-plus-m-squared'", which helps visualize the hypotenuse from origin to tangent point.
• Sign for Angle Bisector: To identify the bisector of the angle containing the origin: Make $C_1, C_2 > 0$. Then $L_1/ \dots = + L_2/ \dots$ gives the bisector containing the origin. The sign of $A_1A_2+B_1B_2$ helps determine acute/obtuse. Think of "Plus for Origin".
• Conic Section Eccentricity:
  • Parabola $\rightarrow e=1$ (Perfect 1)
  • Ellipse $\rightarrow e<1$ (Ess than 1)
  • Hyperbola $\rightarrow e>1$ (Higher than 1)
• Formulas for Conics: For tangent at $(x_1, y_1)$ on the curve, remember the substitution rule: $x^2 \to x x_1$, $y^2 \to y y_1$, $x \to (x+x_1)/2$, $y \to (y+y_1)/2$, $xy \to (xy_1+yx_1)/2$. This rule applies to all second-degree curves (circle, parabola, ellipse, hyperbola) to find tangent, chord of contact, or polar.
• Orthocenter, Centroid, Circumcenter (Euler Line): Old Giant Horror. O, G, H are collinear. G divides OH in ratio 1:2.
• Director Circle Radius: For $x^2+y^2=R^2$, director circle is $x^2+y^2=2R^2$. The radius of the director circle is $\sqrt{2}$ times the radius of the circle itself. This is a common pattern for $a^2+b^2$ vs $a^2-b^2$ compared to just $a^2, b^2$.

IX. Previous-Year Question Patterns

JEE Advanced questions on Coordinate Geometry are often:

1. Multiple Concepts Combined: Expect questions that involve lines and circles, or a conic and a line/circle. For example, finding the equation of a circle tangent to a line and passing through two points, or a tangent to a parabola that passes through a specific point.
2. Locus Problems: Finding the locus of a point satisfying certain geometric conditions. This often involves algebraic manipulation and identifying the resulting curve.
3. Properties of Conics: Questions testing understanding of foci, directrices, eccentricity, latus rectum, tangents, normals, and chords for parabola, ellipse, and hyperbola. Especially their reflective properties.
4. Family of Curves: Using $S_1 + \lambda S_2 = 0$ or $L_1 + \lambda L_2 = 0$ to find specific curves meeting given conditions.
5. Orthocenter, Circumcenter, etc.: Problems involving triangle centers, especially if the vertices are specific points (e.g., vertices of a right-angled triangle).
6. Distance and Angle Calculations: Though seemingly basic, these are often steps within a larger problem.
7. Pair of Lines: Questions on finding the angle between them, or conditions for them to be perpendicular.
8. Common Tangents: Finding common tangents to two circles or a circle and a conic.
9. Homogenization: This is a powerful technique for finding chords of a conic subtending a right angle at the origin, or finding lines joining origin to intersection points of a line and a curve.

Trend Note: Recent JEE Advanced papers tend to focus on conceptual understanding and problem-solving skills rather than rote memorization. Often, questions are framed to test if you can apply multiple concepts synergistically. Derivations of formulas are rarely asked, but their correct application is paramount.

X. Revision Checklist

• Fundamentals:
  • Distance, Section, Area (Triangle), Collinearity.
  • Centroid, Incenter, Circumcenter, Orthocenter formulas and properties.
  • Euler Line.
• Straight Line:
  • Slope in all forms.
  • Equations of lines in all forms (Slope-intercept, Point-slope, Two-point, Intercept, Normal, General).
  • Parallel/Perpendicular conditions.
  • Angle between two lines.
  • Distance of a point from a line, distance between parallel lines.
  • Position of a point W.R.T a line.
  • Family of lines.
  • Reflection of a point across a line.
  • Angle bisectors (acute/obtuse, origin side).
• Pair of Straight Lines:
  • Homogeneous equation $ax^2+2hxy+by^2=0$ (angle, distinct/coincident/imaginary, perpendicular condition).
  • General second-degree equation condition for pair of lines.
  • Point of intersection of pair of lines.
  • Homogenization technique.
• Circle:
  • Standard and General equations.
  • Special cases (passing through origin, touching axes, diameter form).
  • Parametric equation.
  • Position of a point W.R.T a circle, Length of tangent.
  • Tangent equations (at a point, with given slope). Condition of tangency for line $y=mx+c'$.
  • Normal equation.
  • Chord of contact.
  • Director circle.
  • Radical axis, Radical center (properties).
  • Family of circles.
• Parabola:
  • Definition ($e=1$).
  • Standard equations ($y^2=\pm 4ax$, $x^2=\pm 4ay$).
  • Vertex, focus, directrix, axis, latus rectum length and endpoints.
  • Parametric equation.
  • Tangent (at a point, slope form, parametric form). Director circle (directrix).
  • Normal (at a point, slope form, parametric form).
  • Chord of contact.
• Ellipse:
  • Definition ($0<e<1$).
  • Standard equation ($\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$).
  • Center, vertices, co-vertices, foci, directrices.
  • Eccentricity formula ($b^2=a^2(1-e^2)$).
  • Major/Minor axis, Latus Rectum, Auxiliary circle, Eccentric angle.
  • Parametric equation.
  • Tangent (at a point, slope form, parametric form). Director circle.
  • Normal (at a point, parametric form).
  • Chord of contact.
• Hyperbola:
  • Definition ($e>1$).
  • Standard equation ($\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$).
  • Center, vertices, foci, directrices.
  • Eccentricity formula ($b^2=a^2(e^2-1)$).
  • Transverse/Conjugate axis, Latus Rectum.
  • Asymptotes (equations, properties).
  • Parametric equation.
  • Tangent (at a point, slope form, parametric form). Director circle.
  • Normal (at a point, parametric form).
  • Chord of contact.
  • Conjugate Hyperbola, Rectangular Hyperbola ($xy=c^2$).
• Miscellaneous: Pole and Polar concepts (though less frequent, good to know).

XI. Last-Day Strategy

1. Prioritize Weak Areas: Go through your revision checklist one last time. If there are any areas where you feel less confident, quickly review the formulas and solved examples related to them.
2. Formula Scan: Do a rapid scan of all the formulas in this cheat sheet. It's not about memorizing anew, but about refreshing your memory and ensuring instant recall. Focus on the conditions and applications of each formula.
3. Concept Recall: For each section (line, circle, parabola, ellipse, hyperbola), mentally recap the key definitions, properties, and important results (e.g., "Director circle for parabola is its directrix").
4. Don't Learn New Things: Absolutely avoid trying to learn any new concepts or solve complex, unfamiliar problems. This can cause unnecessary anxiety.
5. Sleep Well: A good night's sleep is more beneficial than an extra hour of cramming. A fresh mind helps in applying concepts accurately and avoiding careless errors.
6. Stay Calm and Confident: Trust your preparation. Visualize yourself solving problems accurately. Coordinate Geometry is highly systematic; maintain a logical approach in the exam.
7. Quick Solve Selected PYQs (Optional): If you feel like your mind needs a quick warm-up, briefly look at 2-3 previous year JEE Advanced questions (that you've already solved) and recall their approach. Do not spend more than 30 minutes on this.

Good luck with your JEE Advanced journey!