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Let $3 x - y - 8 = 0$ be the equation of tangent to a parabola at the point $(7, 13)$ . If the focus of the parabola is at $(- 1, - 1)$ , then the equation of its directrix is

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Important Questions on Parabola

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

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?

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The shortest distance between the line

y = x

and the curve

y^{2} = x - 2

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

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:

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 -

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

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 :

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.

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The coordinates of focus of a parabola which touches the lines

x = 0, y = 0, x + y = 1 & y = x - 2

are

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Let a focal chord of parabola

y^{2} = 16 x

cuts it at points

(f, g)

and

(h, k)

. Then

f h

is equal to

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

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

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Find the points on the parabola

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

for which length of the normal is equal to twice of length of the subtangent.

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The set of values of $a$ for which at least one tangent to the parabola $y^{2} = 4 a x$ becomes normal to the circle $x^{2} + y^{2} - 2 a x - 4 a y + 3 a^{2} = 0$ , is (where $a$ is a real number)

Let 3x-y-8=0 be the equation of tangent to a parabola at the point 7,13. If the focus of the parabola is at -1,-1, then the equation of its directrix is

Important Questions on Parabola

Learn and Prepare for any exam you want !

Let $3 x - y - 8 = 0$ be the equation of tangent to a parabola at the point $(7, 13)$ . If the focus of the parabola is at $(- 1, - 1)$ , then the equation of its directrix is