If a x 2 + 2 h x y + b y 2 - 82 x + 98 y + 144 = 0 is the equation of a parabola with focus ( 2 , - 3 ) and directrix 3 x - 2 y + 5 = 0 , then a x 2 + 2 h x y + b y 2 = 0 represents

two lines making an angle π 3 at origin

E M B I B E

Mathematics Crash Course JEE Main>Chapter 20 - Parabola>Level 2>Q 14

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JEE Main

IMPORTANT

Earn 100

The number of points with integral coordinates that lie in the interior of the region common to the circle $x^{2} + y^{2} = 16$ and the parabola $y^{2} = 4 x$ is

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Important Questions on Parabola

MEDIUM

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 20 - Parabola>Level 2>Q 15

A tangent and normal is drawn at the point

P \equiv (16, 16)

of the parabola

y^{2} = 16 x

which cut the axis of the parabola at the points

A

and

B,

respectively. If the centre of the circle through

P,

A

and

B

is

C

, then the angle between

P C

and the axis of

x

is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 20 - Parabola>Level 3>Q 1

The tangent and normal at the point

P (a t^{2}, 2 a t)

to the parabola

y^{2} = 4 a x

meet the

x

-axis in

T

and

G,

respectively, then the angle at which the tangent at

P

to the parabola is inclined to the tangent at

P

to the circle through

P, T, G

is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 20 - Parabola>Level 3>Q 2

Through the vertex

O

of the parabola

y^{2} = 4 a x,

two chords

O P

and

O Q

are drawn and the circles on

O P

and

O Q

as diameters intersect in

R .

If

θ_{1}, θ_{2}

and

ϕ

are the angles made with axis by the tangents at

P

and

Q

on the parabola and by

O R,

then the value of

\cot θ_{1} + \cot θ_{2}

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 20 - Parabola>Level 3>Q 3

A B

is a double ordinate of the parabola

y^{2} = 4 a x .

Tangents drawn to parabola at

A

and

B

meet

y

-axis at

A_{1}

and

B_{1}

, respectively. If the area of trapezium

A A_{1} B_{1} B

is equal to

24 a^{2},

then angle subtended by

A_{1} B_{1}

at the focus of the parabola is equal to

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 20 - Parabola>Level 3>Q 4

Tangent and normal are drawn at

P (16, 16)

on the parabola

y^{2} = 16 x

, which intersect the axis of the parabola at

A

and

B

, respectively. If

C

is the centre of the circle through the points

P, A

and

B

and

∠ C P B = θ

, then the value of

\tan θ

is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 20 - Parabola>Level 3>Q 5

Length of the shortest normal chord of the parabola

y^{2} = 4 a x

is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 20 - Parabola>Level 3>Q 6

The locus of the centre of the circle

C

, which intersect orthogonally and also passing through the focus of the parabola

y^{2} = 4 x

is

MEDIUM

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 20 - Parabola>Level 3>Q 7

All the three vertices of an equilateral triangle lie on the parabola

y = x^{2}

and one of its sides has a slope of

2 .

Then the sum of the

x

-coordinates of the three vertices is

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The number of points with integral coordinates that lie in the interior of the region common to the circle x2+y2=16 and the parabola y2=4x is

Important Questions on Parabola

Learn and Prepare for any exam you want !

The number of points with integral coordinates that lie in the interior of the region common to the circle $x^{2} + y^{2} = 16$ and the parabola $y^{2} = 4 x$ is