## EXERCISE 11.3

#### Page No 255:

#### Question 1:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the *x*-axis, while the minor axis is along the *y*-axis.

On comparing the given equation with, we obtain *a *= 6 and *b* = 4.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (6, 0) and (–6, 0).

Length of major axis = 2*a* = 12

Length of minor axis = 2*b* = 8

Length of latus rectum

#### Question 2:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the *y*-axis, while the minor axis is along the *x*-axis.

On comparing the given equation with, we obtain *b *= 2 and *a* = 5.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, 5) and (0, –5)

Length of major axis = 2*a* = 10

Length of minor axis = 2*b* = 4

Length of latus rectum

#### Question 3:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the *x*-axis, while the minor axis is along the *y*-axis.

On comparing the given equation with, we obtain *a *= 4 and *b* = 3.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are.

Length of major axis = 2*a* = 8

Length of minor axis = 2*b* = 6

Length of latus rectum

#### Question 4:

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the *y*-axis, while the minor axis is along the *x*-axis.

On comparing the given equation with, we obtain *b *= 5 and *a* = 10.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±10).

Length of major axis = 2*a* = 20

Length of minor axis = 2*b* = 10

Length of latus rectum

#### Question 5:

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the *x*-axis, while the minor axis is along the *y*-axis.

On comparing the given equation with, we obtain *a *= 7 and *b* = 6.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (± 7, 0).

Length of major axis = 2*a* = 14

Length of minor axis = 2*b* = 12

Length of latus rectum

#### Question 6:

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the *y*-axis, while the minor axis is along the *x*-axis.

On comparing the given equation with, we obtain *b *= 10 and *a* = 20.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±20)

Length of major axis = 2*a* = 40

Length of minor axis = 2*b* = 20

Length of latus rectum

#### Question 7:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 36*x*^{2} + 4*y*^{2} = 144

#### Answer:

The given equation is 36*x*^{2} + 4*y*^{2} = 144.

It can be written as

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the *y*-axis, while the minor axis is along the *x*-axis.

On comparing equation (1) with, we obtain *b *= 2 and *a* = 6.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±6).

Length of major axis = 2*a *= 12

Length of minor axis = 2*b* = 4

Length of latus rectum

#### Question 8:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 16*x*^{2} + *y*^{2} = 16

#### Answer:

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

It can be written as

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the *y*-axis, while the minor axis is along the *x*-axis.

On comparing equation (1) with, we obtain *b *= 1 and *a* = 4.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±4).

Length of major axis = 2*a* = 8

Length of minor axis = 2*b* = 2

Length of latus rectum

#### Question 9:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 4*x*^{2} + 9*y*^{2} = 36

#### Answer:

The given equation is 4*x*^{2} + 9*y*^{2} = 36.

It can be written as

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the *x*-axis, while the minor axis is along the *y*-axis.

On comparing the given equation with, we obtain *a *= 3 and *b* = 2.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (±3, 0).

Length of major axis = 2*a* = 6

Length of minor axis = 2*b* = 4

Length of latus rectum

#### Question 10:

Find the equation for the ellipse that satisfies the given conditions: Vertices (±5, 0), foci (±4, 0)

#### Answer:

Vertices (±5, 0), foci (±4, 0)

Here, the vertices are on the *x*-axis.

Therefore, the equation of the ellipse will be of the form, where *a *is the semi-major axis.

Accordingly, *a* = 5 and *c* = 4.

It is known that.

Thus, the equation of the ellipse is.

#### Question 11:

Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ±13), foci (0, ±5)

#### Answer:

Vertices (0, ±13), foci (0, ±5)

Here, the vertices are on the *y*-axis.

Therefore, the equation of the ellipse will be of the form, where *a *is the semi-major axis.

Accordingly, *a* = 13 and *c* = 5.

It is known that.

Thus, the equation of the ellipse is.

#### Question 12:

Find the equation for the ellipse that satisfies the given conditions: Vertices (±6, 0), foci (±4, 0)

#### Answer:

Vertices (±6, 0), foci (±4, 0)

Here, the vertices are on the *x*-axis.

Therefore, the equation of the ellipse will be of the form, where *a *is the semi-major axis.

Accordingly, *a* = 6, *c* = 4.

It is known that.

Thus, the equation of the ellipse is.

#### Question 13:

Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (±3, 0), ends of minor axis (0, ±2)

#### Answer:

Ends of major axis (±3, 0), ends of minor axis (0, ±2)

Here, the major axis is along the *x*-axis.

Therefore, the equation of the ellipse will be of the form, where *a *is the semi-major axis.

Accordingly, *a* = 3 and *b* = 2.

Thus, the equation of the ellipse is.

#### Question 14:

Find the equation for the ellipse that satisfies the given conditions: Ends of major axis, ends of minor axis (±1, 0)

#### Answer:

Ends of major axis, ends of minor axis (±1, 0)

Here, the major axis is along the *y*-axis.

Therefore, the equation of the ellipse will be of the form, where *a *is the semi-major axis.

Accordingly, *a* = and *b* = 1.

Thus, the equation of the ellipse is.

#### Question 15:

Find the equation for the ellipse that satisfies the given conditions: Length of major axis 26, foci (±5, 0)

#### Answer:

Length of major axis = 26; foci = (±5, 0).

Since the foci are on the *x*-axis, the major axis is along the *x*-axis.

Therefore, the equation of the ellipse will be of the form, where *a *is the semi-major axis.

Accordingly, 2*a* = 26 ⇒ *a* = 13 and *c* = 5.

It is known that.

Thus, the equation of the ellipse is.

#### Question 16:

Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ±6)

#### Answer:

Length of minor axis = 16; foci = (0, ±6).

Since the foci are on the *y*-axis, the major axis is along the *y*-axis.

Therefore, the equation of the ellipse will be of the form, where *a *is the semi-major axis.

Accordingly, 2*b* = 16 ⇒ *b* = 8 and *c* = 6.

It is known that.

Thus, the equation of the ellipse is.

#### Question 17:

Find the equation for the ellipse that satisfies the given conditions: Foci (±3, 0), *a* = 4

#### Answer:

Foci (±3, 0), *a* = 4

Since the foci are on the *x*-axis, the major axis is along the *x*-axis.

Therefore, the equation of the ellipse will be of the form, where *a *is the semi-major axis.

Accordingly, *c* = 3 and *a* = 4.

It is known that.

Thus, the equation of the ellipse is.

#### Question 18:

Find the equation for the ellipse that satisfies the given conditions: *b* = 3, *c* = 4, centre at the origin; foci on the *x *axis.

#### Answer:

It is given that *b* = 3, *c* = 4, centre at the origin; foci on the *x *axis.

Since the foci are on the *x*-axis, the major axis is along the *x*-axis.

Therefore, the equation of the ellipse will be of the form, where *a *is the semi-major axis.

Accordingly, *b* = 3, *c* = 4.

It is known that.

Thus, the equation of the ellipse is.

#### Question 19:

Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), major axis on the *y*-axis and passes through the points (3, 2) and (1, 6).

#### Answer:

Since the centre is at (0, 0) and the major axis is on the *y*-axis, the equation of the ellipse will be of the form

The ellipse passes through points (3, 2) and (1, 6). Hence,

On solving equations (2) and (3), we obtain *b*^{2} = 10 and *a*^{2} = 40.

Thus, the equation of the ellipse is.

#### Question 20:

Find the equation for the ellipse that satisfies the given conditions: Major axis on the *x*-axis and passes through the points (4, 3) and (6, 2).

#### Answer:

Since the major axis is on the *x*-axis, the equation of the ellipse will be of the form

The ellipse passes through points (4, 3) and (6, 2). Hence,

On solving equations (2) and (3), we obtain *a*^{2} = 52 and *b*^{2} = 13.

Thus, the equation of the ellipse is.