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Parabola

Parabola

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Q.

The cable of a uniformly loaded suspension bridge hangs in the form of a parabola.The roadway which is horizontal and 100 m long is supported by vertical lines wires attached to the cable, the longest wire being 30 m and the shortest wire being 6 m.Find the length of a supporting wire attached to the roadway 18 m from the middle.

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Q. 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

of

P N

, parallel to its axis which meets the parabola at

Q

. If the y-intercept of the line

N Q

is

\frac{4}{3}

, then

$P N = 4$
$M Q = \frac{1}{3}$
$P N = 3$
$M Q = \frac{1}{4}$

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Q.

The equation of the parabola whose focus is $(- 6, - 6)$ and vertex is $(- 2, 2),$ is

$(2 x - y)^{2} - 104 x + 148 y + 124 = 0$
$(2 x + y)^{2} + 104 x + 148 y - 124 = 0$
$(2 x - y)^{2} + 104 x + 148 y - 124 = 0$
$(2 x - y)^{2} + 104 x - 148 y + 124 = 0$

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Q. Axis of a parabola lies along

x

-axis. If its vertex and focus are at distances

2

and

4

respectively from the origin, on the positive

x

-axis then which of the following points does not lie on it?

$(5, 2 \sqrt{6})$
$(8, 6)$
$(6, 4 \sqrt{2})$
$(4, - 4)$

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Q. The equation of the parabola whose vertex and focus lie on the

x -

axis at distances

a

and

a_{1}

(0 < a < a_{1})

from the origin respectively, is

$y^{2} = 4 (a_{1} - a) x$
$y^{2} = 4 (a_{1} - a) (x - a)$
$none of these$
$y^{2} = 4 (a_{1} - a) (x - a_{1})$

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Q. The equation of the parabola whose vertex is at

(3, 2)

and whose focus is at

(5, 2)

$y^{2} - 8 x + 4 y + 12 = 0$
$y^{2} - 8 x - 4 y + 28 = 0$
$y^{2} + 8 x + 4 y - 36 = 0$
$y^{2} - 8 x + 4 y - 28 = 0$

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Q.

Equation of the parabola with its vertex at $(1, 1)$ and focus $(3, 1)$ is

$(x - 1)^{2} = 8 (y - 1)$
${(y - 1)}^{2} = 8 (x - 3)$
${(y - 1)}^{2} = 8 (x - 1)$
${(x - 3)}^{2} = 8 (y - 1)$

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Q. The length of the latus-rectum of the parabola

169 {(x - 1)^{2} + (y - 3)^{2}} = (5 x - 12 y + 17)^{2}

is

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Q. If the circle

x^{2} + y^{2} + 2 λ x = 0, λ \in R

touches the parabola

y^{2} = 4 x

externally, then

$λ > 0$
$λ < 0$
$λ > 1$
$λ < 1$

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Q. The equation of the parabola whose focus is

(4, - 3)

and vertex is

(4, - 1)

$x^{2} - 8 x + 8 y + 24 = 0$
$x^{2} - 8 x - 8 y + 8 = 0$
$x^{2} - 8 x + 8 y + 32 = 0$
$x^{2} - 8 x + 8 y + 8 = 0$

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Q.

Any point on the parabola whose focus is (0, 1) and the directrix is x+2=0 is given by

$t^{2} + 1, 2 t - 1$
$t^{2} + 1, 2 t + 1$
$t^{2}, 2 t$
$t^{2} - 1, 2 t + 1$

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Q. The line

x + 2 y + a = 0

intersects the circle

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

at two distinct points

A

and

B

. Another line

12 x - 6 y - 41 = 0

intersects the circle

x^{2} + y^{2} - 4 x - 2 y + 1 = 0

at two distinct points

C

and

D

. If the four points

A, B, C

, and

D

are concyclic then the value of

a

is

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Q. The radius of the largest circle, which passes through the focus of the parabola

y^{2} = 4 (x + y)

and also contained in it, is

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Q.

Find the equations of the lines joining the vertex of the parabola $y^{2} = 6 x$ to the point on it which have abscissa 24.

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Q. If the parabola

y = a x^{2} + b x + c

, passes through the points

(0, 3), (1, - 4)

and

(- 1, 4)

, then

c - (a + b) =

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Q.

The distance of the point $(1, 1, 9)$ from the point of intersection of the line $(\frac{x - 3}{1}) = (\frac{y - 4}{2}) = (\frac{z - 5}{2})$ and the plane $x + y + z = 17$ is:

$\sqrt{38}$
$19 \sqrt{2}$
$2 \sqrt{19}$
$38$

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Q. The coordinates of an endpoint of the latusrectum of the parabola

(y - 1)^{2} = 4 (x + 1)

are

$(0, 3)$
$(1, 2)$
$(0, - 1)$
$(1, - 2)$

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Q.

At what point of the parabola x^2=9y is the abscissa three times that of ordinate ?

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Q.

Equation of a parabola whose vertex is (2, -3), axis is parallel to the x axis and lastusrectum 8 is

(y+3)²=8(x+2)
(y+3)²=16(x-2)
(y+3)²=32(x-2)
(y+3)²=8(x-2)

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Q.

The locus of the equation $x^{2} + y^{2} = 0$ is

A circle
A hyperbola
A pair of lines
At the origin

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Q. Find the equation of hyperbola whose focus is

(1, 1)

, directrix is

2 x + y = 1

and eccentricity

= \sqrt{3}

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Q.

Using integration, find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4.

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Q. The length of the latusrectum of the parabola

x = 10 y^{2} + b y + c

is

\frac{1}{k}

, then

k =

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Q. Let

O A B C

be a rectangle, where

O

is the origin and

A

,

C

lie on the parabola

y = x^{2}

. If

B

lies on a parabola whose vertex coordinates is

(α, β)

, then the value of

α + β

is

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Q.

Write the coordinates of the orthocentre of the triangle formed by the lines $x^{2} - y^{2} = 0$ and $x + 6 y = 18$ .

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Q. Find the equation of hyperbola whose focus is

(2, - 1)

, directrix is

2 x + 3 y = 1

and eccentricity

= 2

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Q. The coordinates of an endpoint of the latusrectum of the parabola

(y - 1)^{2} = 4 (x + 1)

are

$(0, 3)$
$(1, 2)$
$(0, - 1)$
$(1, - 2)$

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Q.

Any point on the parabola whose focus is (0, 1) and the directrix is x+2=0 is given by

$t^{2} + 1, 2 t - 1$
$t^{2} + 1, 2 t + 1$
$t^{2}, 2 t$
$t^{2} - 1, 2 t + 1$

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Q. The eccentricity of an ellipse with it's centre at origin is

\frac{1}{2}

. If one of the directrices is

x = 8

, then the equation of the ellipse is given by

$3 x^{2} + 4 y^{2} = 48$
$3 x^{2} + 4 y^{2} = 36$
$4 x^{2} + 3 y^{2} = 36$
$4 x^{2} + 3 y^{2} = 48$

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Q. The equation of line(s) joining the vertex of

y^{2} = 6 x

to the point on it whose abscissa is

24,

is (are)

$2 y + x + 1 = 0$
$2 y - x + 1 = 0$
$x + 2 y = 0$
$x - 2 y = 0$

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