Conic Sections Revision Notes for JEE Main

This revision note covers the fundamental concepts, formulas, and problem-solving techniques for Conic Sections, a crucial topic for JEE Main. A thorough understanding of conics is essential for scoring well, as questions often require applying multiple concepts simultaneously.

Key Concepts

• Definition of Conic Sections: A conic section is the locus of a point moving in a plane such that its distance from a fixed point (focus) bears a constant ratio to its distance from a fixed line (directrix). This constant ratio is called the eccentricity, denoted by $e$.
  • If $e = 1$, the conic is a parabola.
  • If $e < 1$, the conic is an ellipse.
  • If $e > 1$, the conic is a hyperbola.
  • If $e = 0$, the conic is a circle (a special case of an ellipse where focus is the center and directrix is at infinity).
• General Equation of a Conic: The general equation of a second-degree curve is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
  • The term $B^2 - 4AC$ determines the type of conic:
    • If $B^2 - 4AC = 0$, it's a parabola.
    • If $B^2 - 4AC < 0$, it's an ellipse (or a circle if $A=C$ and $B=0$).
    • If $B^2 - 4AC > 0$, it's a hyperbola.
    • If $A+C = 0$ (and $B=0$), it's a rectangular hyperbola.
• Parabola:
  • Standard equation: $y^2 = 4ax$.
  • Focus: $(a, 0)$.
  • Directrix: $x = -a$.
  • Axis: $y = 0$ (x-axis).
  • Vertex: $(0, 0)$.
  • Latus Rectum: $4a$.
• Ellipse:
  • Standard equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (major axis along x-axis, $a > b$).
  • Foci: $(\pm ae, 0)$.
  • Directrices: $x = \pm \frac{a}{e}$.
  • Vertices: $(\pm a, 0)$.
  • Minor axis length: $2b$. Distance between foci: $2ae$.
  • Relation between $a, b, e$: $b^2 = a^2(1 - e^2)$.
• Hyperbola:
  • Standard equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.
  • Foci: $(\pm ae, 0)$.
  • Directrices: $x = \pm \frac{a}{e}$.
  • Vertices: $(\pm a, 0)$.
  • Transverse axis length: $2a$. Conjugate axis length: $2b$.
  • Relation between $a, b, e$: $b^2 = a^2(e^2 - 1)$.
  • Asymptotes: $\frac{x}{a} \pm \frac{y}{b} = 0$.
  • Rectangular Hyperbola: A hyperbola where $a=b$. Its asymptotes are perpendicular. $x^2 - y^2 = a^2$ or $xy = c^2$.

Formulas & Results

Parabola

• Equation of a chord joining $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$: $y(t_1 + t_2) = 2x + 2at_1t_2$.
• Equation of tangent at $(x_1, y_1)$: $yy_1 = 2a(x + x_1)$.
• Equation of tangent in terms of slope $m$: $y = mx + \frac{a}{m}$.
• Equation of tangent at $(at^2, 2at)$: $yt = x + at^2$.
• Equation of normal at $(x_1, y_1)$: $y - y_1 = -\frac{y_1}{2a}(x - x_1)$.
• Equation of normal in terms of slope $m$: $y = mx - 2am - am^3$.
• Equation of normal at $(at^2, 2at)$: $y + tx = 2at + at^3$.
• Condition for $y = mx + c$ to be tangent: $c = \frac{a}{m}$.

Ellipse

• Auxiliary Circle: $x^2 + y^2 = a^2$. The eccentric angle $\phi$ is used for parametric form $(a \cos \phi, b \sin \phi)$.
• Equation of tangent at $(x_1, y_1)$: $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$.
• Equation of tangent in terms of slope $m$: $y = mx \pm \sqrt{a^2m^2 + b^2}$.
• Equation of normal at $(x_1, y_1)$: $\frac{a^2x}{x_1} - \frac{b^2y}{y_1} = a^2 - b^2$.
• Equation of chord with midpoint $(x_1, y_1)$: $T = S_1$, i.e., $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} - 1 = \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} - 1$.
• Director Circle: The locus of the point of intersection of perpendicular tangents is $x^2 + y^2 = a^2 + b^2$.

Hyperbola

• Equation of tangent at $(x_1, y_1)$: $\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$.
• Equation of tangent in terms of slope $m$: $y = mx \pm \sqrt{a^2m^2 - b^2}$ (condition $a^2m^2 \ge b^2$).
• Equation of normal at $(x_1, y_1)$: $\frac{a^2x}{x_1} + \frac{b^2y}{y_1} = a^2 + b^2$.
• Equation of chord with midpoint $(x_1, y_1)$: $T = S_1$, i.e., $\frac{xx_1}{a^2} - \frac{yy_1}{b^2} - 1 = \frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} - 1$.
• Director Circle: The locus of the point of intersection of perpendicular tangents is $x^2 + y^2 = a^2 - b^2$.
• Conjugate Hyperbola: $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$. It has the same asymptotes as the original hyperbola.

Worked Examples

Example 1 (Parabola)

Problem: Find the equation of the common tangent to the parabolas $y^2 = 4x$ and $x^2 = 32y$.

Solution: For $y^2 = 4x$, the equation of tangent in slope form is $y = mx + \frac{1}{m}$. (Here $a=1$). For $x^2 = 32y$, its standard form is $x^2 = 4(8)y$. The equation of tangent in slope form is $x = my' + \frac{8}{m'}$ where $y'$ refers to the tangent to $x^2=4ay$. Re-writing in $y=mx+c$ form means $y = \frac{1}{m'}x - \frac{8}{m'^2}$. Let the slope be $m$. So $y = mx - 8m^2$. For a common tangent, the slope $m$ and y-intercept must be equal. So, comparing $y = mx + \frac{1}{m}$ and $y = mx - 8m^2$: $\frac{1}{m} = -8m^2$ $1 = -8m^3$ $m^3 = -\frac{1}{8} \Rightarrow m = -\frac{1}{2}$. Substitute $m = -\frac{1}{2}$ into $y = mx + \frac{1}{m}$: $y = -\frac{1}{2}x + \frac{1}{(-1/2)}$ $y = -\frac{1}{2}x - 2$ $2y = -x - 4$ $x + 2y + 4 = 0$. This is the equation of the common tangent.

Example 2 (Ellipse)

Problem: An ellipse passes through the point $(4, \sqrt{6})$ and has foci $(\pm 2, 0)$. Find its eccentricity.

Solution: Since the foci are $(\pm 2, 0)$, the major axis is along the x-axis. The distance from the center to a focus is $c = ae = 2$. The standard equation of the ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. We know $b^2 = a^2(1 - e^2)$. Also, $ae=2 \Rightarrow e = 2/a$. So, $b^2 = a^2(1 - (2/a)^2) = a^2(1 - 4/a^2) = a^2 - 4$. The equation becomes $\frac{x^2}{a^2} + \frac{y^2}{a^2 - 4} = 1$. The ellipse passes through $(4, \sqrt{6})$. Substitute these coordinates: $\frac{4^2}{a^2} + \frac{(\sqrt{6})^2}{a^2 - 4} = 1$ $\frac{16}{a^2} + \frac{6}{a^2 - 4} = 1$. Multiply by $a^2(a^2 - 4)$: $16(a^2 - 4) + 6a^2 = a^2(a^2 - 4)$ $16a^2 - 64 + 6a^2 = a^4 - 4a^2$ $22a^2 - 64 = a^4 - 4a^2$ $a^4 - 26a^2 + 64 = 0$. This is a quadratic equation in $a^2$. Let $X = a^2$. $X^2 - 26X + 64 = 0$. Using the quadratic formula, $X = \frac{-(-26) \pm \sqrt{(-26)^2 - 4(1)(64)}}{2(1)}$ $X = \frac{26 \pm \sqrt{676 - 256}}{2}$ $X = \frac{26 \pm \sqrt{420}}{2} = \frac{26 \pm 2\sqrt{105}}{2} = 13 \pm \sqrt{105}$. Since $a^2 > 4$ (because $b^2 = a^2 - 4 > 0$), we must have $a^2 = 13 + \sqrt{105}$. Now find eccentricity $e$. We know $ae = 2 \Rightarrow e = 2/a$. $e^2 = \frac{4}{a^2} = \frac{4}{13 + \sqrt{105}}$. Rationalize the denominator: $e^2 = \frac{4}{13 + \sqrt{105}} \times \frac{13 - \sqrt{105}}{13 - \sqrt{105}} = \frac{4(13 - \sqrt{105})}{169 - 105} = \frac{4(13 - \sqrt{105})}{64} = \frac{13 - \sqrt{105}}{16}$. So $e = \sqrt{\frac{13 - \sqrt{105}}{16}}$.

Example 3 (Hyperbola)

Problem: If the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{b^2} = 1$ and the hyperbola $\frac{x^2}{144} - \frac{y^2}{81} = \frac{1}{25}$ coincide, then find the value of $b^2$.

Solution: For the ellipse $\frac{x^2}{16} + \frac{y^2}{b^2} = 1$, we have $A_E^2 = 16$. Foci are $(\pm A_E e_E, 0)$. Let $c_E$ be the distance from center to focus. $c_E^2 = A_E^2 - b^2 = 16 - b^2$. So, the foci are $(\pm \sqrt{16-b^2}, 0)$.

For the hyperbola $\frac{x^2}{144} - \frac{y^2}{81} = \frac{1}{25}$, first standardize it: $\frac{x^2}{144/25} - \frac{y^2}{81/25} = 1$. Here, $A_H^2 = \frac{144}{25}$ and $B_H^2 = \frac{81}{25}$. Let $c_H$ be the distance from center to focus. $c_H^2 = A_H^2 + B_H^2 = \frac{144}{25} + \frac{81}{25} = \frac{225}{25} = 9$. So, $c_H = \sqrt{9} = 3$. The foci of the hyperbola are $(\pm 3, 0)$.

Since the foci coincide, we must have: $\sqrt{16 - b^2} = 3$. Square both sides: $16 - b^2 = 9$. $b^2 = 16 - 9$. $b^2 = 7$.

Common Mistakes

• Confusing $a$ and $b$ in Ellipse/Hyperbola: In an ellipse, $a$ refers to the semi-major axis ($a > b$), not necessarily the denominator under $x^2$. For $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$, if $A > B$, then $a=A$. If $B > A$, then $a=B$, and the major axis is along the y-axis. In a hyperbola, $a^2$ is always the denominator of the positive term.
• Incorrect Eccentricity Formulas: For ellipse, $b^2 = a^2(1 - e^2)$. For hyperbola, $b^2 = a^2(e^2 - 1)$. Be careful with the signs.
• Sign Errors in Tangent/Normal Equations: Especially in hyperbola, positive/negative signs change compared to an ellipse (e.g., $a^2+b^2$ vs $a^2-b^2$ in director circles).
• Forgetting Conditions for Tangency: For $y=mx+c$ to be tangent to $\frac{x^2}{a^2} \pm \frac{y^2}{b^2} = 1$, specific conditions apply. For hyperbola, $a^2m^2 \ge b^2$ is sometimes important.
• Not Standardizing Equations: Always convert the given equation to its standard form before applying formulas (e.g., dividing by constants, completing the square for shifted conics).
• Mixing up Asymptotes and Directrices: Asymptotes are lines that the hyperbola approaches at infinity. Directrices are related to the definition of a conic.
• Parabola with Axis Parallel to Y-axis: For $x^2 = 4ay$, the focus is $(0, a)$, directrix $y=-a$, and tangent is $x = my - am^2$. Don't use formulas for $y^2 = 4ax$ directly.
• Incorrectly Identifying Conic from General Equation: The condition $B^2 - 4AC$ helps identify the conic, but cases like $A=C, B=0$ imply a circle. Degenerate conics (pairs of lines, points) are also possible.

Quick Revision Checklist

• Definition & Eccentricity: Understand $e$ and how it defines the type of conic.
• Standard Equations: Memorise the standard forms for parabola ($y^2=4ax, x^2=4ay$), ellipse ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$), and hyperbola ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$).
• Key Features: Know how to find foci, directrices, vertices, axes, and length of latus rectum for all standard conics.
• Eccentricity Relations: $b^2 = a^2(1-e^2)$ for ellipse, $b^2 = a^2(e^2-1)$ for hyperbola.
• Tangent/Normal Equations: Be proficient with point form $(x_1, y_1)$ and slope form ($m$) for all conics.
• Chord Equations: Especially $T=S_1$ for chord with given midpoint.
• Director Circle: Know the equations for ellipse ($x^2+y^2=a^2+b^2$) and hyperbola ($x^2+y^2=a^2-b^2$).
• Parametric Coordinates: $(at^2, 2at)$ for parabola, $(a \cos\phi, b \sin\phi)$ for ellipse, $(a \sec\theta, b \tan\theta)$ for hyperbola.
• Asymptotes of Hyperbola: $y = \pm \frac{b}{a}x$.
• Rectangular Hyperbola: $a=b$, asymptotes perpendicular, equation $xy=c^2$.
• General Second-Degree Equation: Use $B^2-4AC$ to identify conic type.
• Shifted Conics: Be able to translate the origin to find features of conics like $(y-k)^2 = 4a(x-h)$.