HARD

Earn 100

Three distinct normals to the parabola $y^{2} = x$ are drawn through a point $(c, 0)$ , then

50% studentsanswered this correctly

Important Questions on Parabola

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The shortest distance between the point

(\frac{3}{2}, 0)

and the curve

y = \sqrt{x}, (x > 0)

, 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 :

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The shortest distance between the parabolas

y^{2} = 4 x

and

y^{2} = 2 x - 6

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

Normals drawn to

y^{2} = 4 a x

at the points where it is intersected by the line

y = m x + c

intersected at

P

. Coordinates of foot of the another normal drawn to the parabola from the point

P

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 equation of the normal to the parabola

y^{2} = 4 x

which is perpendicular to

x + 3 y + 1 = 0

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

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-

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

Three normals are drawn from the point

(c, 0)

to the curve

y^{2} = x

. If one of the normals is

X

-axis, then the value of

c

for which the other two normals are perpendicular to each other is

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

P

and

Q

are two distinct points on the parabola,

y^{2} = 4 x

, with parameters

t

and

t_{1}

, respectively. If the normal at

P

passes through

Q

, then the minimum value of

t_{1}^{2}

, is

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The normal at a point on the parabola

y^{2} = 4 x

passes through

(5, 0)

. If two more normals to this parabola also pass through

(5, 0),

then the centroid of the triangle formed by the feet of these three normals is

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The slopes of the normals to the parabola

y^{2} = 4 a x

intersecting at a point on the axis of the parabola at a distance

4 a

from its vertex are in

EASY

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The normal to the parabola

y^{2} = 8 x

at the point

(2, 4)

meets the parabola again at the point

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 =

Three distinct normals to the parabola y2=x are drawn through a point c, 0, then

Important Questions on Parabola

Learn and Prepare for any exam you want !

Three distinct normals to the parabola $y^{2} = x$ are drawn through a point $(c, 0)$ , then