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The equation of the mirror that can reflect all incident rays from origin parallel to $y$ -axis is

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Important Questions on Parabola

EASY

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

If the length of the latus rectum of an ellipse is

4

units and the distance between a focus and its nearest vertex on the major axis is

\frac{3}{2}

units, then its eccentricity is

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Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Let

S

and

S^{'}

be the foci of an ellipse and

B

be any one of the extremities of its minor axis. If

Δ S^{'} B S

is a right angled triangle with right angle at

B

and area

(Δ S^{'} B S) = 8 sq . units

, then the length of a latus rectum of the ellipse is :

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is:

EASY

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

A focus of an ellipse is at the origin. The directrix is the line

x = 4

and the eccentricity is

1 / 2 .

Then the length of the semi-major axis is

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

If a point

P (x, y)

moves along the ellipse

\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1

and if

C

is the centre of the ellipse, then the sum of maximum and minimum values of

C P

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Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The equation of the circle passing through the foci of the ellipse

\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1

, and having centre at

(0, 3)

EASY

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

5 x + 9 = 0

is the directrix of the hyperbola

16 x^{2} - 9 y^{2} = 144,

then its corresponding focus is:

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Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The focus of the parabola

y = 2 x^{2} + x

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Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus ractum to the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1,

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 :

EASY

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

If the eccentricity of the standard hyperbola passing through the point

(4,6)

2,

then the equation of the tangent to the hyperbola at

(4,6)

is:

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Consider an ellipse, whose center is at the origin and its major axis is along the

x

-axis. If its eccentricity is

\frac{3}{5}

and the distance between its foci is

6

, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:

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Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at

(0,5 \sqrt{3}),

then the length of its latus rectum is:

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Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

If the line

x - 2 y = 12

is a tangent to the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

at the point

(3, - \frac{9}{2})

, then the length of the latus rectum of the ellipse is

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Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

If a directrix of a hyperbola centered at the origin and passing through the point

(4, - 2 \sqrt{3})

5 x = 4 \sqrt{5}

and its eccentricity is

e

, then:

EASY

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

A hyperbola has its centre at the origin, passes through the point

(4, 2)

and has transverse axis of length

4

along the

x - a x i s .

Then the eccentricity of the hyperbola is:

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 =

The equation of the mirror that can reflect all incident rays from origin parallel to y-axis is

Important Questions on Parabola

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The equation of the mirror that can reflect all incident rays from origin parallel to $y$ -axis is