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Locus of point of intersection of the straight lines (where $t$ is a real variable parameter) $x - (t + 1) y - (t - 1) = 0 & x - t y - 2 (t - 1) = 0$ is

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Important Questions on Parabola

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

Consider the parabola

y^{2} = 4 x

. Let

P

and

Q

be points on the parabola where

P (4, - 4)

and

Q (9, 6) .

Let

R

be a point on the arc of the parabola between

P

and

Q .

Then, the area of

Δ P Q R

is largest when

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

Let

A, B

be two distinct points on the parabola

y^{2} = 4 x

. If the axis of the parabola touches a circle of radius

r

having

A B

as diameter, the slope of the line

A B

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

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:

EASY

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The curved described parametrically by

x = t^{2} + t + 1

and

y = t^{2} - t + 1

represents:

HARD

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

For the parabola

y^{2} + 6 y - 2 x + 5 = 0,

match the items in List-

I

with the suitable item in List-

II

given below:

List- $I$		List- $II$
$(I)$	Vertex	$(A)$	$(- \frac{3}{2}, - 3)$
$(II)$	Focus	$(B)$	$(\frac{3}{2}, - 3)$
$(III)$	Equation of the directrix	$(C)$	$2 x + 5 = 0$
$(IV)$	Equation of the axis	$(D)$	$2 x + y + 3 = 0$
		$(E)$	$y + 3 = 0$
		$(F)$	$(- 2, - 3)$

The correct matching is

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Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

Match the items given in List-A with those of the items of List-B

	List-A		List-B
(A)	The vertex of the parabola $y^{2} + 4 x - 2 y + 3 = 0$ is	(I)	$(\frac{5}{4}, 1)$
(B)	The vertex of the parabola $x^{2} + 8 x + 12 y + 4 = 0$ is	(II)	$(1, \frac{5}{4})$
(C)	The focus of the parabola $y^{2} - x - 2 y + 2 = 0$ is	(III)	$(\frac{- 1}{2}, 1)$
(D)	The focus of the parabola $x^{2} - 2 x - 8 y - 23 = 0$ is	(IV)	$(1, - 1)$
		(V)	$(- 4, 1)$

The correct match is

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

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

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 focus of the parabola

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

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

Let

P

be a variable point on the parabola

y = 4 x^{2} + 1 .

Then, the locus of the mid-point of the point

P

and the foot of the perpendicular drawn from the point

P

to the line

y = x

is:

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The locus of the mid-point of the line segment joining the focus of the parabola

y^{2} = 4 a x

to a moving point of the parabola, is another parabola whose directrix is:

EASY

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The parametric equations of the parabola

y^{2} - 4 x - 8 y - 12 = 0

are

EASY

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

A point on the parabola whose focus and vertex are respectively at

(\frac{5}{4}, - 2)

and

(1, - 2)

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Basics of Parabola

The coordinates of the focus of the parabola described parametrically by

x = 5 t^{2} + 2, y =

10 t + 4

(where

t

is a parameter) are

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

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

Locus of point of intersection of the straight lines (where t is a real variable parameter) x-t+1y-t-1=0 & x-ty-2t-1=0is

Important Questions on Parabola

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Locus of point of intersection of the straight lines (where $t$ is a real variable parameter) $x - (t + 1) y - (t - 1) = 0 & x - t y - 2 (t - 1) = 0$ is