HARD

JEE Advanced

IMPORTANT

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Let $E$ denote the parabola $y^{2} = 8 x .$ Let $P = (- 2, 4)$ , and let $Q$ and $Q^{'}$ be two distinct points on $E$ such that the lines $P Q$ and $P Q^{'}$ are tangents to $E$ . Let $F$ be the focus of $E$ . Then which of the following statements is (are) TRUE?

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Important Questions on Parabola

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 15 - Parabola>JEE Advanced Paper 2 - 2019>Q 3

Let the circles

C_{1} : x^{2} + y^{2} = 9

and

C_{2} : {(x - 3)}^{2} + {(y - 4)}^{2} = 16,

intersect at the points

X

and

Y .

Suppose that another circle

C_{3} : {(x - h)}^{2} + {(y - k)}^{2} = r^{2}

satisfies the following conditions:

(i)

centre of

C_{3}

is collinear with the centres of

C_{1}

and

C_{2}

(i i) C_{1}

and

C_{2}

both lie inside

C_{3},

and

(i i i) C_{3}

touches

C_{1}

M

and

C_{2}

N .

Let the line through

X

and

Y

intersect

C_{3}

Z

and

W,

and let a common tangent of

C_{1}

and

C_{3}

be a tangent to the parabola

x^{2} = 8 α y .

There are some expression given in the List-

I

whose values are given in List-

I I

below:

	List- I		List- II
$(I)$	$2 h + k$	$(P)$	$6$
$(I I)$	$\frac{L e n g t h o f Z W}{L e n g t h o f X Y}$	$(Q)$	$\sqrt{6}$
$(I I I)$	$\frac{A r e a o f t r i a n g l e M Z N}{A r e a o f t r i a n g l e Z M W}$	$(R)$	$\frac{5}{4}$
$(I V)$	$α$	$(S)$	$\frac{21}{5}$
		$(T)$	$2 \sqrt{6}$
		$(U)$	$\frac{10}{3}$

Which of the following is the only INCORRECT combination?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 15 - Parabola>JEE Advanced Paper 2 - 2019>Q 4

Let the circles

C_{1} : x^{2} + y^{2} = 9

and

C_{2} : {(x - 3)}^{2} + {(y - 4)}^{2} = 16,

intersect at the points

X

and

Y .

Suppose that another circle

C_{3} : {(x - h)}^{2} + {(y - k)}^{2} = r^{2}

satisfies the following conditions:

(i)

centre of

C_{3}

is collinear with the centres of

C_{1}

and

C_{2}

(i i) C_{1}

and

C_{2}

both lie inside

C_{3},

and

(i i i) C_{3}

touches

C_{1}

M

and

C_{2}

N .

Let the line through

X

and

Y

intersect

C_{3}

Z

and

W,

and let a common tangent of

C_{1}

and

C_{3}

be a tangent to the parabola

x^{2} = 8 α y .

There are some expression given in the List-

I

whose values are given in List-

I I

below:

	List- $I$		List- II
$(I)$	$2 h + k$	$(P)$	$6$
$(I I)$	$\frac{L e n g t h o f Z W}{L e n g t h o f X Y}$	$(Q)$	$\sqrt{6}$
$(I I I)$	$\frac{A r e a o f t r i a n g l e M Z N}{A r e a o f t r i a n g l e Z M W}$	$(R)$	$\frac{5}{4}$
$(I V)$	$α$	$(S)$	$\frac{21}{5}$
		$(T)$	$2 \sqrt{6}$
		$(U)$	$\frac{10}{3}$

Which of the following is the only INCORRECT combination?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 15 - Parabola>JEE Advanced Paper 2 - 2016>Q 5

Let

P

be the point on the parabola

y^{2} = 4 x

which is at the shortest distance from the center

S

of the circle

x^{2} + y^{2} - 4 x - 16 y + 64 = 0 .

Let

Q

be the point on the circle dividing the line segment

S P

internally. Then -

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 15 - Parabola>JEE Advanced Paper 2 - 2014>Q 7

The common tangents to the circle

x^{2} + y^{2} = 2

and the parabola

y^{2} = 8 x

touch the circle at the points

P,

Q

and the parabola at the points

R, S

. Then the area of the quadrilateral

S R P Q

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 15 - Parabola>JEE Advanced Paper 1 - 2017>Q 11

If a chord, which is not a tangent, of the parabola

y^{2} = 16 x

has the equation

2 x + y = p

, and midpoint

(h, k)

, then which of the following is (are) possible value(s) of

p, h

and

k

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 15 - Parabola>JEE Advanced Paper 1 - 2017>Q 12

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

If a tangent to a suitable conic (Column 1) is found to be

y = x + 8

and its point of contact is

(8, 16)

, then which of the following options is the only Correct combination?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 15 - Parabola>JEE Advanced Paper 1 - 2015>Q 15

Let

P

and

Q

be distinct points on the parabola

y^{2} = 2 x

such that a circle with

P Q

as diameter passes through the vertex

O

of the parabola. If

P

lies in the first quadrant and the area of the triangle

Δ O P Q

3 \sqrt{2}

sq. units, then which of the following is (are) the coordinates of

P

Let E denote the parabola y2=8x. Let P=-2,4, and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E. Let F be the focus of E. Then which of the following statements is (are) TRUE?

Important Questions on Parabola

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Let $E$ denote the parabola $y^{2} = 8 x .$ Let $P = (- 2, 4)$ , and let $Q$ and $Q^{'}$ be two distinct points on $E$ such that the lines $P Q$ and $P Q^{'}$ are tangents to $E$ . Let $F$ be the focus of $E$ . Then which of the following statements is (are) TRUE?