Conic Sections

Conic sections are the curves you get when you slice a double cone with a plane. Depending on the angle of the slice, you can create a circle, a parabola, an ellipse, or a hyperbola. This chapter is a detailed study of these four curves using coordinate geometry. You will learn to recognize each conic from its standard second-degree equation and understand its fundamental properties like focus, directrix, and eccentricity, which define the shape and orientation of the curve.

The chapter is typically divided into the study of the parabola, the ellipse, and the hyperbola in sequence. For each, you will derive their standard equations and learn to find key features such as the vertex, axes, latus rectum, and asymptotes (for the hyperbola). A significant part of the chapter involves finding the equations of tangents and normals to these curves at a given point or from an external point. You will also work with related concepts like the chord of contact and the director circle.

The main challenge in Conic Sections is the large number of formulas and standard results that need to be remembered and applied correctly under pressure. Problems can become complex when they involve finding the locus of a point or combine properties of tangents and normals with geometric conditions. A solid understanding of earlier chapters like Straight Lines and Circles is necessary, as many concepts from those topics are used extensively here.

Formulas for Conic Sections

General Equation of a Second Degree: $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ Represents different conics based on the values of the coefficients.
Condition for a Pair of Straight Lines: $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$ The condition for the general second-degree equation to represent two straight lines.
Eccentricity (e): $e = \frac{SP}{PM}$ SP is the distance from a point on the conic to the focus; PM is its distance to the directrix.
Standard Equation of a Parabola: $y^2 = 4ax$ Vertex at (0,0), focus at (a,0), and directrix x = -a. Eccentricity e = 1.
Parametric Form of a Parabola: $x = at^2, y = 2at$ Any point on the parabola y^2 = 4ax can be represented as (at^2, 2at).
Tangent to Parabola y² = 4ax at (x₁, y₁): $yy_1 = 2a(x + x_1)$ This is the point form of the tangent equation.
Standard Equation of an Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ For a > b, eccentricity e is given by b² = a²(1 - e²). For an ellipse, 0 < e < 1.
Tangent to Ellipse in Slope Form: $y = mx \pm \sqrt{a^2m^2 + b^2}$ These are the two tangents to the standard ellipse with a given slope m.
Standard Equation of a Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Eccentricity e is given by b² = a²(e² - 1). For a hyperbola, e > 1.
Asymptotes of a Hyperbola: $\frac{x}{a} \pm \frac{y}{b} = 0$ These are the pair of lines that the hyperbola approaches at infinity.
Equation of Tangent at (x₁, y₁): $T = 0$ A shorthand for replacing x²→xx₁, y²→yy₁, 2xy→xy₁+yx₁, 2x→x+x₁, 2y→y+y₁ in the conic's equation.
Equation of Chord of Contact: $T = 0$ The line joining points of contact of tangents drawn from an external point (x₁, y₁). The formula is identical to the tangent equation.

One of these questions, worked through

JEE Advanced 2023

Let $C_1$ be the circle of radius 1 with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A=(4,1)$ , where $1<r<3$ . Two distinct common tangents $PQ$ and $ST$ of $C_1$ and $C_2$ are drawn. The tangent $PQ$ touches $C_1$ at $P$ and $C_2$ at $Q$ . The tangent $ST$ touches $C_1$ at $S$ and $C_2$ at $T$ . Mid points of the line segments $PQ$ and $ST$ are joined to form a line which meets the $x$ -axis at a point $B$ . If $AB = \sqrt{5}$ , then the value of $r^2$ is

Conic Sections Let

C_1

be the circle of radius 1 with center at the origin. Let

C_2

be the circle of radius

r

with center at the point

A=(4,1)

, where

1<r<3

. Two distinct common tangents

PQ

and

ST

C_1

and

C_2

are drawn. The tangent

PQ

touches

C_1

P

and

C_2

Q

. The tangent

ST

touches

C_1

S

and

C_2

T

. Mid points of the line segments

PQ

and

ST

are joined to form a line which meets the

x

-axis at a point

B

. If

AB = \sqrt{5}

, then the value of

r^2

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Let C₁ be the circle of radius 1 with center at the origin. Let C₂ be the circle of radius r with center at the point A=(4,1), where 1<r<3. Two distinct common tangents PQ and ST of C₁ and C₂ are drawn. The tangent PQ touches C₁ at P and C₂ at Q. The tangent ST touches C₁ at S and C₂ at T. Mid points of the line segments PQ and ST are joined to form a line which meets the x-axis at a point B. If AB = √{5}, then the value of r² is $\text{Given: } C_1:\ \text{center } O(0,0),\ \text{radius } 1$ $C_2:\ \text{center } A(4,1),\ \text{radius } r,\quad 1 < r < 3$ $\text{common tangents } PQ,\ ST \Rightarrow \text{midpoints joined meet } x\text{-axis at } B$ $\text{To find: } r^2,\ \text{given } AB = \sqrt{5}$ $1.\ \text{draw the circles and their common tangents}$ $2.\ \text{tangent } x\cos\theta + y\sin\theta = 1$ $3.\ \text{line of midpoints, then } B, \text{ then } r^2$ $C_1:\ x^2 + y^2 = 1$ $C_2:\ (x-4)^2 + (y-1)^2 = r^2$ $OA = \sqrt{4^2 + 1^2} = \sqrt{17}$ $\text{two direct common tangents } PQ,\ ST$ $\text{tangent to } C_1:\ x\cos\theta + y\sin\theta = 1$ $\text{point of contact } P = (\cos\theta,\ \sin\theta)$ $\left|\dfrac{4\cos\theta + \sin\theta - 1}{\sqrt{\cos^2\theta + \sin^2\theta}}\right| = r$ $\Rightarrow\ 4\cos\theta + \sin\theta = 1 - r$ $Q = A + r(\cos\theta,\sin\theta) = (4 + r\cos\theta,\ 1 + r\sin\theta)$ $M = \left(\dfrac{4 + (1+r)\cos\theta}{2},\ \dfrac{1 + (1+r)\sin\theta}{2}\right)$ $4M_x + M_y = \dfrac{17 + (1+r)(4\cos\theta + \sin\theta)}{2}$ $= \dfrac{17 + (1+r)(1-r)}{2} = \dfrac{18 - r^2}{2}$ $\text{line of midpoints}:\ 4x + y = \dfrac{18 - r^2}{2}$ $y = 0:\ 4x = \dfrac{18 - r^2}{2} \Rightarrow x = \dfrac{18 - r^2}{8}$ $B = \left(\dfrac{18 - r^2}{8},\ 0\right)$ $AB^2 = \left(4 - \dfrac{18 - r^2}{8}\right)^2 + 1 = 5$ $\Rightarrow\ \left(4 - \dfrac{18 - r^2}{8}\right)^2 = 4$ $4 - \dfrac{18 - r^2}{8} = \pm 2$ $+2:\ 18 - r^2 = 16 \ \Rightarrow \ r^2 = 2$ $-2:\ 18 - r^2 = 48 \ \Rightarrow \ r^2 = -30 \ \text{(rejected)}$ $\boxed{r^2 = 2}$

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Past-paper questions on Conic Sections

JEE Advanced 69 questions