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Let $P$ be a point, different from $(0, 0)$ , on the parabola $y^{2} = 4 x$ whose focus is at $F$ . Let $T$ be the foot of the perpendicular drawn from $P$ on to the directrix of the parabola. The least distance of the point $(0, 0)$ from the orthocentre of triangle $P F T$ as $P$ moves on the parabola is

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Important Questions on Parabola

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The centres of those circles which touch the circle,

x^{2} + y^{2} - 8 x - 8 y - 4 = 0,

externally and also touch the

x

- axis, lie on

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

y = m x + c

is the normal at a point on the parabola

y^{2} = 8 x

whose focal distance is

8

units, then

|c|

is equal to:

HARD

Mathematics>Coordinate Geometry>Parabola>Basics 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

EASY

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The vertex of the parabola

y = x^{2} - 2 x + 4

is shifted

p

units to the right and then

q

units up. If the resulting point is

(4, 5),

then the values of

p

and

q

respectively are

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

P Q

be a double ordinate of the parabola,

y^{2} = - 4 x,

where

P

lies in the second quadrant. If

R

divides

P Q

in the ratio

2 : 1,

then the locus of

R

is:

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The equation of the directrix of the parabola

x^{2} - 4 x - 3 y + 10 = 0

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

A chord is drawn through the focus of the parabola

y^{2} = 6 x

such that its distance from the vertex of this parabola is

\frac{\sqrt{5}}{2}

, then its slope can be

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The directrix of the parabola

2 y^{2} + 25 x = 0

is _________

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

Suppose the parabola

{(y - k)}^{2} = 4 (x - h),

with vertex

A

, passes through

O = (0, 0)

and

L = (0, 2) .

Let

D

be an end point of the latus rectum. Let the

y

-axis intersect the axis of the parabola at

P

. Then

∠ P D A

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The focus of the parabola

y^{2} - 4 y - x + 3 = 0

EASY

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The vertex of the parabola

y = (x - 2) (x - 8) + 7

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

Let

P (4, - 4)

and

Q (9, 6)

be two points on the parabola,

y^{2} = 4 x

and let

X

be any point on the arc

P O Q

of this parabola, where

O

is the vertex of this parabola, such that the area of

∆ P X Q

is maximum. Then this maximum area (in sq. units) is :

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The area (in sq. units) of an equilateral triangle inscribed in the parabola

y^{2} = 8 x,

with one of its vertices on the vertex of this parabola is

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

If one end of a focal chord of the parabola,

y^{2} = 16 x

is at

(1, 4)

, then the length of this focal chord is

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The equation

y^{2} + 3 = 2 (2 x + y)

represents a parabola with the vertex at

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The length of latus rectum of the parabola whose focus is at

(1, - 2)

and directrix is the line

x + y + 3 = 0

EASY

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

If the parabola

x^{2} = 4 a y

passes through the point

(2, 1)

, then the length of the latus rectum is

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The focus of the parabola

{(y + 1)}^{2} = - 8 (x + 2)

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

Let

A (4, - 4)

and

B (9, 6)

be points on the parabola,

y^{2} = 4 x .

Let

C

be chosen on the arc

A O B

of the parabola, where

O

is the origin, such that the area of

Δ A C B

is maximum. Then, the area (in sq. units) of

Δ A C B

, is:

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

Let

O

be the vertex and

Q

be any point on the parabola,

x^{2} = 8 y

. If the point

P

divides the line segment

O Q

internally in the ratio

1 : 3

, then the locus of

P

Let P be a point, different from (0,0), on the parabola y2=4x whose focus is at F. Let T be the foot of the perpendicular drawn from P on to the directrix of the parabola. The least distance of the point (0,0) from the orthocentre of triangle PFT as P moves on the parabola is

Important Questions on Parabola

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