Given : A circle, 2 x 2 + 2 y 2 = 5 and a parabola, y 2 = 4 5 x . Statement - I : An equation of a common tangent to these curves is y = x + 5 . Statement - II : If the line, y = m x + 5 m m ≠ 0 is their common tangent, then m satisfies m 4 - 3 m 2 + 2 = 0 .

Statement - I is true; Statement - II is true; Statement - II is not a correct explanation for statement - I .

Statement - I is true; Statement - II is false.

Statement - I is false; Statement - II is true.

Statement - I is true; Statement - II is true; Statement - II is a correct explanation for statement - I .

Given : A circle, 2 x 2 + 2 y 2 = 5 and a parabola, y 2 = 4 5 x . Statement - I : An equation of a common tangent to these curves is y = x + 5 . Statement - II : If the line, y = m x + 5 m m ≠ 0 is their common tangent, then m satisfies m 4 - 3 m 2 + 2 = 0 .

Statement - I is true; Statement - II is true; Statement - II is not a correct explanation for statement - I .

HARD

JEE Main

IMPORTANT

Earn 100

Given : A circle, $2 x^{2} + 2 y^{2} = 5$ and a parabola, $y^{2} = 4 \sqrt{5} x$ .

Statement - I : An equation of a common tangent to these curves is $y = x + \sqrt{5}$ .

Statement - II : If the line, $y = m x + \frac{\sqrt{5}}{m} (m \neq 0)$ is their common tangent, then $m$ satisfies $m^{4} - 3 m^{2} + 2 = 0$ .

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Important Questions on Parabola

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 16 - Parabola>JEE Main - 24 June 2022 Shift 1>Q 53

If two tangents drawn from a point

(α, β)

lying on the ellipse

25 x^{2} + 4 y^{2} = 1

to the parabola

y^{2} = 4 x

are such that the slope of one tangent is four times the other, then the value of

{(10 α + 5)}^{2} + {(16 β^{2} + 50)}^{2}

equals ______

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 16 - Parabola>JEE Main - 24 June 2022 Shift 2>Q 54

Let

P_{1}

be a parabola with vertex

(3, 2)

and focus

(4, 4)

and

P_{2}

be its mirror image with respect to the line

x + 2 y = 6

. Then the directrix of

P_{2}

x + 2 y =

_____.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 16 - Parabola>JEE Main - 25 June 2022 Shift 1>Q 55

y = m_{1} x + c_{1}

and

y = m_{2} x + c_{2}, m_{1} \neq m_{2}

are two common tangents of circle

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

and parabola

y^{2} = x

, then the value of

8 |m_{1} m_{2}|

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 16 - Parabola>JEE Main - 25 June 2022 Shift 1>Q 56

Let

x = 2 t, y = \frac{t^{2}}{3}

be a conic. Let

S

be the focus and

B

be the point on the axis of the conic such that

S A ⊥ B A

, where

A

is any point on the conic. If

k

is the ordinate of the centroid of the

Δ S A B

, then

\lim_{t \to 1} k

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 16 - Parabola>JEE Main - 26 June 2022 Shift 1>Q 57

Let the normal at the point

P

on the parabola

y^{2} = 6 x

pass through the point

(5, - 8)

. If the tangent at

P

to the parabola intersects its directrix at the point

Q

, then the ordinate of the point

Q

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 16 - Parabola>JEE Main - 26 June 2022 Shift 1>Q 58

Let the common tangents to the curves

4 (x^{2} + y^{2}) = 9

and

y^{2} = 4 x

intersect at the point

Q

. Let an ellipse, centered at the origin

O

, has lengths of semi-minor and semi-major axes equal to

O Q

and

6

, respectively. If

e

and

l

respectively denote the eccentricity and the length of the latus rectum of this ellipse, then

\frac{l}{e^{2}}

is equal to ______.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 16 - Parabola>JEE Main - 27 June 2022 Shift 1>Q 59

A circle of radius

2

unit passes through the vertex and the focus of the parabola

y^{2} = 2 x

and touches the parabola

y = {(x - \frac{1}{4})}^{2} + α

, where

α > 0

. Then

{(4 α - 8)}^{2}

is equal to ______.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 16 - Parabola>JEE Main - 27 June 2022 Shift 2>Q 60

If the equation of the parabola, whose vertex is at

(5, 4)

and the directrix is

3 x + y - 29 = 0

, is

x^{2} + a y^{2} + b x y + c x + d y + k = 0

, then

a + b + c + d + k

is equal to

Given : A circle, 2x2+2y2=5 and a parabola, y2=45x. Statement - I : An equation of a common tangent to these curves is y=x+5. Statement - II : If the line, y=mx+5m m≠0 is their common tangent, then m satisfies m4-3m2+2=0.

Important Questions on Parabola

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