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Conic Sections

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 2010

LEVELJEE Advanced

Comprehension Passage

Tangents are drawn from the point

P (3, 4)

to the ellipse

\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1

touching the ellipse at points

A

and

B

Question 1:

The coordinates of $A$ and $B$ are

(A)

(3, 0)

and

(0, 2)

(B)

(\frac{8}{5}, \frac{2 161}{15})

and

(\frac{9}{5}, \frac{8}{5})

(C)

(- \frac{8}{5}, \frac{2 161}{15})

and

(0, 2)

(D)

(3, 0)

and

(- \frac{9}{5}, \frac{8}{5})

Question 2:

The orthocenter of the triangle $P A B$ is

(A)

(5, \frac{8}{7})

(B)

(\frac{7}{5}, \frac{25}{8})

(C)

(\frac{11}{5}, \frac{8}{5})

(D)

(\frac{8}{25}, \frac{7}{5})

Question 3:

The equation of the locus of the point whose distances from the point $P$ and the line $A B$ are equal, is

(A)

9 x^{2} + y^{2} - 6 x y - 54 x - 62 y + 241 = 0

(B)

x^{2} + 9 y^{2} + 6 x y - 54 x + 62 y - 241 = 0

(C)

9 x^{2} + 9 y^{2} - 6 x y - 54 x - 62 y - 241 = 0

(D)

x^{2} + y^{2} - 2 x y + 27 x + 31 y - 120 = 0

JEE Advanced 2023

LEVELJEE Advanced

Let $T_{1}$ and $T_{2}$ be two distinct common tangents to the ellipse $E : \frac{x ^{2}}{6} + \frac{y ^{2}}{3} = 1$ and the parabola $P : y^{2} = 12 x$ . Suppose that the tangent $T_{1}$ touches $P$ and $E$ at the point $A_{1}$ and $A_{2}$ , respectively and the tangent $T_{2}$ touches $P$ and $E$ at the points $A_{4}$ and $A_{3}$ , respectively. Then which of the following statements is(are) true?

* Multiple Correct Options

(A)

The area of the quadrilateral

A_{1} A_{2} A_{3} A_{4}

is 35 square units

(B)

The area of the quadrilateral

A_{1} A_{2} A_{3} A_{4}

is 36 square units

(C)

The tangents

T_{1}

and

T_{2}

meet the x-axis at the point

(- 3, 0)

(D)

The tangents

T_{1}

and

T_{2}

meet the x-axis at the point

(- 6, 0)

JEE Advanced 2022

LEVELJEE Advanced

Consider the parabola $y^{2} = 4 x$ . Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = (- 2, 1)$ meet the parabola at $P_{1}$ and $P_{2}$ . Let $Q_{1}$ and $Q_{2}$ be points on the lines $S P_{1}$ and $S P_{2}$ respectively such that $P Q_{1}$ is perpendicular to $S P_{1}$ and $P Q_{2}$ is perpendicular to $S P_{2}$ . Then, which of the following is/are TRUE ?

* Multiple Correct Options

(A)

S Q_{1} = 2

(B)

Q_{1} Q_{2} = \frac{3 10}{5}

(C)

P Q_{1} = 3

(D)

S Q_{2} = 1

JEE Advanced 2021

LEVELJEE Advanced

Comprehension Passage

Consider the lines

L_{1}

and

L_{2}

defined by

L_{1} : x 2 + y - 1 = 0

and

L_{2} : x 2 - y + 1 = 0

. For a fixed constant

λ

, let

C

be the locus of a point

P

such that the product of the distance of

P

from

L_{1}

and the distance of

P

from

L_{2}

λ^{2}

. The line

y = 2 x + 1

meets

C

at two points

R

and

S

, where the distance between

R

and

S

270

. Let the perpendicular bisector of

R S

meet

C

at two distinct points

R^{'}

and

S^{'}

. Let

D

be the square of the distance between

R^{'}

and

S^{'}

Question 1:

The value of $λ^{2}$ is

Question 2:

The value of $D$ is

JEE Advanced 2020

LEVELJEE Advanced

Let $a$ and $b$ be positive real numbers such that $a > 1$ and $b < a$ . Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ . Suppose the tangent to the hyperbola at $P$ passes through the point $(1, 0)$ , and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $Δ$ denote the area of the triangle formed by the tangent at $P$ , the normal at $P$ and the $x$ -axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

* Multiple Correct Options

(A)

1 < e < 2

(B)

2 < e < 2

(C)

Δ = a^{4}

(D)

Δ = b^{4}

JEE Main 2025 (January)

LEVELJEE Advanced

Let the ellipse $E_{1} : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ $a > b$ and $E_{2} : \frac{x ^{2}}{A ^{2}} + \frac{y ^{2}}{B ^{2}} = 1$ , $A < B$ have same eccentricity $\frac{1}{3}$ . Let the product of their lengths of latus rectums be $\frac{32}{3}$ , and the distance between the foci of $E_{1}$ be 4. If $E_{1}$ and $E_{2}$ meet at A, B, C and D, then the area of the quadrilateral ABCD equals :

(A)

\frac{12 6}{5}

(B)

66

(C)

\frac{18 6}{5}

(D)

\frac{24 6}{5}

JEE Main 2025 (January)

LEVELJEE Advanced

Let A and B be the two points of intersection of the line $y + 5 = 0$ and the mirror image of the parabola $y^{2} = 4 x$ with respect to the line $x + y + 4 = 0$ If d denotes the distance between A and B, and a denotes the area of $△ S A B,$ where S is the focus of the parabola $y^{2} = 4 x$ then the value of $(a + d)$ is

JEE Main 2025 April

LEVELJEE Advanced

Let the sum of the focal distances of the point $P (4, 3)$ on the hyperbola $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be $\frac{8 5}{3}$ . If for $H$ , the length of the latus rectum is $l$ and the product of the focal distances of the point $P$ is $m$ , then $9 l^{2} + 6 m$ is equal to :-

JEE Main 2024 (09 Apr Shift 2)

LEVELJEE Advanced

Consider the circle $C : x^{2} + y^{2} = 4$ and the parabola $P : y^{2} = 8 x$ . If the set of all values of $α$ , for which three chords of the circle $C$ on three distinct lines passing through the point $(α, 0)$ are bisected by the parabola $P$ is the interval $(p, q)$ , then $(2 q - p)^{2}$ is equal to ________

JEE Main 2023 (24 January Shift 2)

LEVELJEE Advanced

The equations of the sides AB and AC of a triangle ABC are $(λ + 1) x + λ y = 4$ and $λ x + (1 - λ) y + λ = 0$ respectively. Its vertex A is on the y-axis and its orthocentre is $(1, 2)$ . The length of the tangent from the point C to the part of the parabola $y^{2} = 6 x$ in the first quadrant is

(A)

6

(B)

22

(C)

(D)

JEE Main 2022 (29 July Shift 1)

LEVELJEE Advanced

Let the focal chord of the parabola $P : y^{2} = 4 x$ along the line $L : y = m x + c, m > 0$ meet the parabola at the points $M$ and $N$ . Let the line $L$ be a tangent to the hyperbola $H : x^{2} - y^{2} = 4$ . If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive x-axis, then the area of the quadrilateral $O M F N$ is :

(A)

26

(B)

214

(C)

46

(D)

414

JEE Main 2021 (18 March Shift 1)

LEVELJEE Advanced

A square ABCD has all its vertices on the curve $x^{2} y^{2} = 1$ . The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is

JEE Advanced 2006

LEVELJEE Advanced

Comprehension Passage

A B C D

is a square of side length 2 units.

C_{1}

is the circle touching all the sides of the square

A B C D

and

C_{2}

is the circumcircle of square

A B C D

L

is a fixed line in the same plane and

R

is a fixed point.

Question 1:

If $P$ is any point of $C_{1}$ and $Q$ is another point on $C_{2}$ , then $\frac{P A ^{2} + P B ^{2} + P C ^{2} + P D ^{2}}{Q A ^{2} + Q B ^{2} + Q C ^{2} + Q D ^{2}}$ is equal to

If a circle is such that it touches the line $L$ and the circle $C_{1}$ externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is

pair of straight line

Question 3:

A line $L^{'}$ through $A$ is drawn parallel to $B D$ . Point $S$ moves such that its distances from the line $B D$ and the vertex $A$ are equal. If locus of $S$ cuts $L^{'}$ at $T_{2}$ and $T_{3}$ and $A C$ at $T_{1}$ , then area of $Δ T_{1} T_{2} T_{3}$ is

A vertical line passing through the point $(h, 0)$ intersects the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{3} = 1$ at the points $P$ and $Q$ . Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$ . If $Δ (h) = area of the triangle P QR$ , $Δ_{1} = max_{1/2 \leq h \leq 1} Δ (h)$ and $Δ_{2} = min_{1/2 \leq h \leq 1} Δ (h)$ , then $\frac{8}{5} Δ_{1} - 8 Δ_{2} =$

JEE Advanced 2016

LEVELJEE Advanced

Comprehension Passage

Let

F_{1} (x_{1}, 0)

and

F_{2} (x_{2}, 0)

for

x_{1} < 0

and

x_{2} > 0

, be the foci of the ellipse

\frac{x ^{2}}{9} + \frac{y ^{2}}{8} = 1

. Suppose a parabola having vertex at the origin and focus at

F_{2}

intersects the ellipse at point

M

in the first quadrant and at point

N

in the fourth quadrant.

Question 1:

The orthocentre of the triangle $F_{1} M N$ is

(A)

(- \frac{9}{10}, 0)

(B)

(\frac{2}{3}, 0)

(C)

(\frac{9}{10}, 0)

(D)

(\frac{2}{3}, 6)

Question 2:

If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to the parabola at $M$ meets the x-axis at $Q$ , then the ratio of area of the triangle $M QR$ to area of the quadrilateral $M F_{1} N F_{2}$ is

Comprehension Passage

The circle

x^{2} + y^{2} - 8 x = 0

and hyperbola

\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1

intersect at the points

A

and

B

Question 1:

Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

(A)

2 x - 5 y - 20 = 0

(B)

2 x - 5 y + 4 = 0

(C)

3 x - 4 y + 8 = 0

(D)

4 x - 3 y + 4 = 0

Question 2:

Equation of the circle with $A B$ as its diameter is

(A)

x^{2} + y^{2} - 12 x + 24 = 0

(B)

x^{2} + y^{2} + 12 x + 24 = 0

(C)

x^{2} + y^{2} + 24 x - 12 = 0

(D)

x^{2} + y^{2} - 24 x - 12 = 0

JEE Advanced 2000

LEVELJEE Advanced

Let $A B C$ be an equilateral triangle inscribed in the circle $x^{2} + y^{2} = a^{2}$ . Suppose perpendiculars from $A, B, C$ to the major axis of the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, (a > b)$ meets the ellipse respectively, at $P, Q, R$ , so that $P, Q, R$ lie on the same side of the major axis as $A, B, C$ respectively. Prove that the normals to the ellipse drawn at the points $P, Q$ and $R$ are concurrent.

JEE Main 2021 (27 July Shift 1)

LEVELJEE Advanced

A ray of light through $(2, 1)$ is reflected at a point $P$ on the $y$ -axis and then passes through the point $(5, 3)$ . If this reflected ray is the directrix of an ellipse with eccentricity $\frac{1}{3}$ and the distance of the nearer focus from this directrix is $\frac{8}{53}$ , then the equation of the other directrix can be:

(A)

11 x + 7 y + 8 = 0

11 x + 7 y - 15 = 0

(B)

11 x - 7 y - 8 = 0

11 x + 7 y + 15 = 0

(C)

2 x - 7 y + 29 = 0

2 x - 7 y - 7 = 0

(D)

2 x - 7 y - 39 = 0

2 x - 7 y - 7 = 0

JEE Main 2023 (08 Apr Shift 2)

LEVELJEE Advanced

Let $A (0, 1)$ , $B (1, 1)$ and $C (1, 0)$ be the mid-points of the sides of a triangle with incentre at the point $D$ . If the focus of the parabola $y^{2} = 4 a x$ passing through $D$ is $(α + β 2, 0)$ , where $α$ and $β$ are rational numbers, then $\frac{α}{β ^{2}}$ is equal to

\frac{9}{2}

JEE Advanced 2018

LEVELJEE Advanced

Let $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ , where $a > b > 0$ , be a hyperbola in the $x y$ -plane whose conjugate axis $L M$ subtends an angle of $6 0^{\circ}$ at one of its vertices $N$ . Let the area of the triangle $L M N$ be $43$ . Match the properties in List-I to the numbers in List-II.

List-I

(P)

The length of the conjugate axis of

H

(Q)

The eccentricity of

H

(R)

The distance between the foci of

H

(S)

The length of the latus rectum of

H

List-II

(1)

(2)

(3)

\frac{2}{3}

(4)

43

Micro-Topic Strategy Matrix

Standard Equations of Parabola, Ellipse, and Hyperbola

Foci, Directrices, and Eccentricity

Parametric Equations

Equation of Tangent and Normal

Chord in terms of Midpoint

Asymptotes of a Hyperbola

Rectangular Hyperbola

Chapter Elite-20 Challenge Zone

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coordinates of A and B are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The orthocenter of the triangle PAB is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the locus of the point whose distances from the point P and the line AB are equal, is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of λ2 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of D is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A square ABCD has all its vertices on the curve x2y2=1. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P is any point of C1​ and Q is another point on C2​, then QA2+QB2+QC2+QD2PA2+PB2+PC2+PD2​ is equal to

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The orthocentre of the triangle F1​MN is

Comprehension Passage

Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Equation of the circle with AB as its diameter is

List-I

List-II

The coordinates of $A$ and $B$ are

The orthocenter of the triangle $P A B$ is

The equation of the locus of the point whose distances from the point $P$ and the line $A B$ are equal, is

The value of $λ^{2}$ is

The value of $D$ is

A square ABCD has all its vertices on the curve $x^{2} y^{2} = 1$ . The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is

If $P$ is any point of $C_{1}$ and $Q$ is another point on $C_{2}$ , then $\frac{P A ^{2} + P B ^{2} + P C ^{2} + P D ^{2}}{Q A ^{2} + Q B ^{2} + Q C ^{2} + Q D ^{2}}$ is equal to

The orthocentre of the triangle $F_{1} M N$ is

Equation of the circle with $A B$ as its diameter is