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If the normals drawn at the points $t_{1}$ and $t_{2}$ on the parabola meet the parabola again at its point $t_{3}$ , then $t_{1} t_{2}$ equals

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Important Questions on Parabola

HARD

Mathematics>Coordinate Geometry>Parabola>Normals to a 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?

HARD

Mathematics>Coordinate Geometry>Parabola>Normals to a 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>Normals to a Parabola

The shortest distance between the parabolas

y^{2} = 4 x

and

y^{2} = 2 x - 6

HARD

Mathematics>Coordinate Geometry>Parabola>Normals to a 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

HARD

Mathematics>Coordinate Geometry>Parabola>Normals to a 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>Normals to a 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

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals to a 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

EASY

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

Consider the parabola with vertex

(\frac{1}{2}, \frac{3}{4})

and the directrix

y = \frac{1}{2} .

Let

P

be the point where the parabola meets the line

x = - \frac{1}{2}

. If the normal to the parabola at

P

intersects the parabola again at the point

Q .

then

(PQ)^{2}

is equal to :

HARD

Mathematics>Coordinate Geometry>Parabola>Normals to a 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>Normals to a Parabola

The condition that the line

\frac{x}{p} + \frac{y}{q} = 1

be a normal to the parabola

y^{2} = 4 a x

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals to a 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>Normals to a 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 =

EASY

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

The normal to the parabola

y^{2} = 8 x

at the point

(2, 4)

meets the parabola again at the point

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

The focus of the parabola

y = 2 x^{2} + x

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals to a 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

HARD

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

Let

P

be a point on the parabola

y^{2} = 4 a x

, where

a > 0

. The normal to the parabola at

P

meets the

x

-axis at a point

Q

. The area of the triangle

P F Q

, where

F

is the focus of the parabola, is

120

. If the slope

m

of the normal and

a

are both positive integers, then the pair

(a, m)

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

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

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

The shortest distance between the point

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

and the curve

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

, is

EASY

Mathematics>Coordinate Geometry>Parabola>Normals to a 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>Normals to a 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-

If the normals drawn at the points t1 and t2 on the parabola meet the parabola again at its point t3, then t1t2 equals

Important Questions on Parabola

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If the normals drawn at the points $t_{1}$ and $t_{2}$ on the parabola meet the parabola again at its point $t_{3}$ , then $t_{1} t_{2}$ equals