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If $P (a t^{2}, 2 a t)$ be one end of a focal chord of the parabola $y^{2} = 4 a x,$ then the length of the chord is

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Important Questions on Parabola

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The vertices of the base of an isosceles triangle lie on a parabola

y^{2} = 4 x

and the base is a part of the line

y = 2 x - 4

. If the third vertex of the triangle lies on the

x

-axis, its coordinates are

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

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 & B

, respectively. If

C

is the center of the circle through the points

P, A & B

and

∠ C P B = θ,

then a value of

\tan θ

is:

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Let

P

be a point on the parabola,

y^{2} = 12 x

and

N

be the foot of the perpendicular drawn from

P

, on the axis of the parabola. A line is now drawn through the mid-point

M

P N

, parallel to its axis which meets the parabola at

Q

. If the

y -

intercept of the line

NQ

\frac{4}{3},

then :

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Suppose

O A B C

is a rectangle in the

x y

-plane where

O

is the origin and

A, B

lie on the parabola

y = x^{2} .

Then

C

must lie on the curve-

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

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?

EASY

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The line

y = 4 x + c

touches the parabola

y^{2} = 4 x

, if

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

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

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The shortest distance between the line

y = x

and the curve

y^{2} = x - 2

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The focus of the parabola

y = 2 x^{2} + x

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

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 -

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Let

C

be the locus of the mirror image of a point on the parabola

y^{2} = 4 x

with respect to the line

y = x

. Then the equation of tangent to

C

P (2, 1)

is :

EASY

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

If the three normals drawn to the parabola,

y^{2} = 2 x

pass through the point

(a, 0), a \neq 0

, then

a

must be greater than :

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

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

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The length of the chord of the parabola

y^{2} = 4 a x (a > 0)

which passes through the vertex and makes an acute angle

α

with the axis of the parabola is

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

If two tangents to the parabola

y^{2} = 8 x

meet the tangent at its vertex in

M

and

N

such that

M N = 4

, then the locus of the point of intersection of those two tangents is

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The shortest distance between the line

x - y = 1

and the curve

x^{2} = 2 y

is:

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The length of the chord of the parabola

x^{2} = 4 y

having equation

x - \sqrt{2} y + 4 \sqrt{2} = 0

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

If the tangents and normals at the extremities of a focal chord of a parabola intersect at

(x_{1}, y_{1})

and

(x_{2}, y_{2})

respectively, then

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Let

A (1, 2), B (4, - 4), C (2, 2 \sqrt{2})

be points on the parabola

y^{2} = 4 x .

α

and

β

respectively represent the area of

Δ A B C

and the area of the triangle formed by the tangents at

A, B, C

to the above parabola, then

α β =

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Prove that the semi-latus rectum is a harmonic mean between the segments of any focal chord of a parabola.

If Pat2,2at be one end of a focal chord of the parabola y2=4ax, then the length of the chord is

Important Questions on Parabola

Learn and Prepare for any exam you want !

If $P (a t^{2}, 2 a t)$ be one end of a focal chord of the parabola $y^{2} = 4 a x,$ then the length of the chord is