## EXERCISE 11.2

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#### Question 1:

Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for *y*^{2} = 12*x*

#### Answer:

The given equation is *y*^{2} = 12*x*.

Here, the coefficient of *x *is positive. Hence, the parabola opens towards the right.

On comparing this equation with *y*^{2 }= 4*ax*, we obtain

4*a* = 12 ⇒ *a* = 3

∴Coordinates of the focus = (*a*, 0) = (3, 0)

Since the given equation involves *y*^{2}, the axis of the parabola is the *x*-axis.

Equation of direcctrix, *x* = –*a *i.e., *x *= – 3 i.e., *x* + 3 = 0

Length of latus rectum = 4*a* = 4 × 3 = 12

#### Question 2:

Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for *x*^{2} = 6*y*

#### Answer:

The given equation is *x*^{2} = 6*y*.

Here, the coefficient of *y *is positive. Hence, the parabola opens upwards.

On comparing this equation with *x*^{2} = 4*ay*, we obtain

∴Coordinates of the focus = (0, *a*) =

Since the given equation involves *x*^{2}, the axis of the parabola is the *y*-axis.

Equation of directrix,

Length of latus rectum = 4*a* = 6

#### Question 3:

Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for *y*^{2} = – 8*x*

#### Answer:

The given equation is *y*^{2} = –8*x*.

Here, the coefficient of *x *is negative. Hence, the parabola opens towards the left.

On comparing this equation with *y*^{2} = –4*ax*, we obtain

–4*a* = –8 ⇒ *a* = 2

∴Coordinates of the focus = (–*a*, 0) = (–2, 0)

Since the given equation involves *y*^{2}, the axis of the parabola is the *x*-axis.

Equation of directrix, *x* = *a* i.e.,* x* = 2

Length of latus rectum = 4*a* = 8

#### Question 4:

Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for *x*^{2} = – 16*y*

#### Answer:

The given equation is *x*^{2} = –16*y*.

Here, the coefficient of *y *is negative. Hence, the parabola opens downwards.

On comparing this equation with *x*^{2}* = – *4*ay, *we obtain

–4*a* = –16 ⇒ *a* = 4

∴Coordinates of the focus = (0, –*a*) = (0, –4)

Since the given equation involves *x*^{2}, the axis of the parabola is the *y*-axis.

Equation of directrix, *y* = *a* i.e., *y* = 4

Length of latus rectum = 4*a* = 16

#### Question 5:

Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for *y*^{2} = 10*x*

#### Answer:

The given equation is *y*^{2} = 10*x*.

Here, the coefficient of *x *is positive. Hence, the parabola opens towards the right.

On comparing this equation with *y*^{2 }= 4*ax*, we obtain

∴Coordinates of the focus = (*a*, 0)

Since the given equation involves *y*^{2}, the axis of the parabola is the *x*-axis.

Equation of directrix,

Length of latus rectum = 4*a* = 10

#### Question 6:

Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for *x*^{2} = –9*y*

#### Answer:

The given equation is *x*^{2} = –9*y*.

Here, the coefficient of *y *is negative. Hence, the parabola opens downwards.

On comparing this equation with *x*^{2} = –4*ay*, we obtain

∴Coordinates of the focus =

Since the given equation involves *x*^{2}, the axis of the parabola is the *y*-axis.

Equation of directrix,

Length of latus rectum = 4*a* = 9

#### Page No 247:

#### Question 7:

Find the equation of the parabola that satisfies the following conditions: Focus (6, 0); directrix *x* = –6

#### Answer:

Focus (6, 0); directrix, *x* = –6

Since the focus lies on the *x*-axis, the *x-*axis is the axis of the parabola.

Therefore, the equation of the parabola is either of the form *y*^{2} = 4*ax* or

*y*^{2} = – 4*ax*.

It is also seen that the directrix, *x* = –6 is to the left of the *y*-axis, while the focus (6, 0) is to the right of the *y*-axis. Hence, the parabola is of the form *y*^{2} = 4*ax*.

Here, *a* = 6

Thus, the equation of the parabola is *y*^{2} = 24*x*.

#### Question 8:

Find the equation of the parabola that satisfies the following conditions: Focus (0, –3); directrix *y* = 3

#### Answer:

Focus = (0, –3); directrix *y* = 3

Since the focus lies on the *y*-axis, the *y-*axis is the axis of the parabola.

Therefore, the equation of the parabola is either of the form *x*^{2} = 4*ay* or

*x*^{2 }= – 4*ay*.

It is also seen that the directrix, *y* = 3 is above the *x*-axis, while the focus

(0, –3) is below the *x*-axis. Hence, the parabola is of the form *x*^{2} = –4*ay*.

Here, *a* = 3

Thus, the equation of the parabola is *x*^{2} = –12*y*.

#### Question 9:

Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0); focus (3, 0)

#### Answer:

Vertex (0, 0); focus (3, 0)

Since the vertex of the parabola is (0, 0) and the focus lies on the positive *x*-axis, *x*-axis is the axis of the parabola, while the equation of the parabola is of the form *y*^{2} = 4*ax*.

Since the focus is (3, 0), *a* = 3.

Thus, the equation of the parabola is *y*^{2} = 4 × 3 × *x*, i.e., *y*^{2} = 12*x*

#### Question 10:

Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0) focus (–2, 0)

#### Answer:

Vertex (0, 0) focus (–2, 0)

Since the vertex of the parabola is (0, 0) and the focus lies on the negative *x*-axis, *x*-axis is the axis of the parabola, while the equation of the parabola is of the form *y*^{2} = –4*ax*.

Since the focus is (–2, 0), *a* = 2.

Thus, the equation of the parabola is *y*^{2} = –4(2)*x*, i.e., *y*^{2} = –8*x*

#### Question 11:

Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0) passing through (2, 3) and axis is along *x*-axis

#### Answer:

Since the vertex is (0, 0) and the axis of the parabola is the *x*-axis, the equation of the parabola is either of the form *y*^{2} = 4*ax* or *y*^{2} = –4*ax*.

The parabola passes through point (2, 3), which lies in the first quadrant.

Therefore, the equation of the parabola is of the form *y*^{2} = 4*ax*, while point

(2, 3) must satisfy the equation *y*^{2} = 4*ax*.

Thus, the equation of the parabola is

#### Question 12:

Find the equation of the parabola that satisfies the following conditions: Vertex (0, 0), passing through (5, 2) and symmetric with respect to *y*-axis

#### Answer:

Since the vertex is (0, 0) and the parabola is symmetric about the *y*-axis, the equation of the parabola is either of the form *x*^{2} = 4*ay* or *x*^{2} = –4*ay*.

The parabola passes through point (5, 2), which lies in the first quadrant.

Therefore, the equation of the parabola is of the form *x*^{2} = 4*ay*, while point

(5, 2) must satisfy the equation *x*^{2} = 4*ay*.

Thus, the equation of the parabola is