# Class 11 Maths NCERT Solutions for Chapter 11 Conic Sections Exercise 11.3

### Conic Sections Exercise 11.3 Solutions

**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 x ^{2}/36 + y^{2}/16 = 1 **

**Solution**

The given equation is x^{2}/36 + y^{2}/16 = 1.

Here, the denominator of x^{2}/36 is greater than the denominator of y^{2}/16 .

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

On comparing the given equation with x^{2}/a^{2} + y^{2}/b^{2} = 1 , we obtain a = 6 and b = 4.

Therefore,

The coordinates of the foci are (2√5, 0) and (-2√5, 0) .

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

Length of major axis = 2a = 12

Length of minor axis = 2b = 8

**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 x ^{2}/4 + y^{2}/25 = 1 **

**Solution**

The given equation is

Here, the denominator of y^{2} /25 is greater than the denominator of x^{2} /4.

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 (0, √21) and (0, - √21).

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

Length of major axis = 2a = 10

Length of minor axis = 2b = 4

**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 x ^{2}/16 + y^{2}/9 = 1.**

**Solution**

The given equation is

Here, the denominator of x^{2} /16 is greater than the denominator of y^{2} /9 .

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

on comparing the given equation with x^{2}/a^{2} + y^{2}/b^{2} = 1, we obtain a = 4 and b = 3.

Therefore,

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

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

Length of major axis = 2a = 8

Length of minor axis = 2b = 6

**4. 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 x ^{2}/25 + y^{2}/100 = 1 **

**Solution**

The given equation is

Here, the denominator of y^{2}/100 is greater than the denominator of x^{2}/25.

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

On comparing the given equation with x^{2} /b^{2} + y^{2} /a^{2} = 1, we obtain b = 5 and a = 10.

Therefore,

The coordinates of the foci are (0, ± 5√3) .

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

Length of major axis = 2a = 20

Length of minor axis = 2b = 10

**5. 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 x ^{2}/49 + y^{2}/36 = 1 **

**Solution**

The given equation is

Here, the denominator of x^{2}/49 is greater than the denominator of y^{2}/36.

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

On comparing the given equation with x^{2}/a^{2} + y^{2}/b^{2} = 1, we obtain a = 7 and b = 6.

Therefore,

The coordinates of the foci are (± √13, 0) .

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

Length of major axis = 2a = 14

Length of minor axis = 2b = 12

**6. 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 x ^{2}/100 + y^{2}/400 = 1**

**Solution**

The given equation is

Here, the denominator of y^{2} /400 is greater than the denominator of x^{2}/100.

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

On comparing the given equation with x^{2}/b^{2} + y^{2}/a^{2} = 1, we obtain b = 10 and a = 20.

Therefore,

The coordinates of the foci are (0, ± 10√3).

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

Length of major axis = 2a = 40

Length of minor axis = 2b = 20

**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 36x ^{2} + 4y^{2} = 144**

**Solution**

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

It can be written as

Here, the denominator of y^{2}/6^{2} is greater than the denominator of x^{2}/2^{2}.

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

On comparing equation (1) with x^{2}/b^{2} + y^{2}/a^{2} = 1, we obtain b = 2 and a = 6.

Therefore,

The coordinates of the foci are (0, ± 4√2).

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

Length of major axis = 2a = 12

Length of minor axis = 2b = 4

**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 16x ^{2} + y^{2} = 16**

**Solution**

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

It can be written as

Here, the denominator of y^{2}/4^{2} is greater than the denominator of x^{2}/1^{2}.

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

On comparing equation (1) with x^{2}/b^{2} + y^{2}/a^{2} = 1, we obtain b = 1 and a = 4.

Therefore,

The coordinates of the foci are (0, ± √15).

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

length of major axis = 2a = 8

Length of minor axis = 2b = 2

Eccentricity, e = c/a = √15/4

Length of latus rectum - 2b^{2}/a = (2×1)/4 = 1/2

**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 4x ^{2} + 9y^{2} = 36**

**Solution**

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

It can be written as

Here, the denominator of x^{2}/3^{2} is greater than the denominator of y^{2}/2^{2} .

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

On comparing the given equation with x^{2}/a^{2} + y^{2}/b^{2} = , we obtain a = 3 and b = 2.

Therefore,

The coordinates of the foci are (±√5, 0).

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

Length of major axis = 2a = 6

Length of minor axis = 2b = 4

Eccentricity, e = c/a = √5/3

Length of latus rectum = 2b^{2}/a = (2× 4)/3 = 8/3

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

**Solution**

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 x^{2}/a^{2} + y^{2}/b^{2} = 1 , where *a *is the semi-major axis.

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

It is known that a^{2} = b^{2} + c^{2}.

∴ 5^{2} = b^{2} + 4^{2}

⇒ 25 = b^{2} + 16

⇒ b^{2} = 25 - 16

⇒ b = √9 = 3

Thus, the equation of the ellipse

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

**Solution**

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 x^{2} /a^{2} + y^{2} /b^{2} = 1, where *a *is the semi-major axis.

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

It is known that a^{2} = b^{2} + c^{2}.

∴ 13^{2} = b^{2} + 5^{2}

⇒ 169 = b^{2} + 25

⇒ b^{2} = 169 - 25

⇒ b = √144 = 12

Thus, the equation of the ellipse is

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

**Solution**

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 x^{2} /a^{2} + y^{2} /b^{2} = 1, where *a *is the semi-major axis.

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

It is known that a^{2} = b^{2} + c^{2}.

∴ 6^{2} = b^{2} + 4^{2}

⇒ 36 = b^{2} + 16

⇒ b^{2} = 36 - 16

⇒ b = √ 20

Thus, the equation of the ellipse is

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

**Solution**

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 x^{2}/a^{2} + y^{2}/b^{2} = 1, where a is the semi-major axis.

Accordingly, a = 3 and b = 2.

Thus, the equation of the ellipse is

**14. Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (0, ± √5), ends of minor axis (±1, 0)**

**Solution**

Ends of major axis (0, ± √5), 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 x^{2} /a^{2} + y^{2} /b^{2} = 1, where *a *is the semi-major axis.

Accordingly, *a* = √5 and *b* = 1.

Thus, the equation of the ellipse is

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

**Solution**

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 x^{2} /a^{2} + y^{2} /b^{2} = 1, where *a *is the semi-major axis.

Accordingly,

2*a* = 26

⇒ *a* = 13 and *c* = 5.

It is known that a^{2} = b^{2} + c^{2}.

∴ 13^{2} = b^{2} + 5^{2}

⇒ 169 = b^{2} + 25

⇒ b^{2} = 169 - 25

⇒ b = √144 = 12

Thus, the equation of the ellipse is

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

**Solution**

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 x^{2} /a^{2} + y^{2} /b^{2} = 1, where a is the semi-major axis.

Accordingly,

2b = 16

⇒ b = 8 and c = 6.

It is known that a^{2} = b^{2} + c^{2}.

∴ a^{2} = 8^{2} + 6^{2} = 64 + 36 = 100

⇒ a = √100 = 10

Thus, the equation of the ellipse is

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

**Solution**

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 x^{2}/a^{2} + y^{2}/b^{2} = 1, where a is the semi-major axis.

Accordingly, c = 3 and a = 4.

It is known that a^{2} = b^{2} + c^{2}.

∴ 4^{2} = b^{2} + 3^{2}

⇒ 16 = b^{2} + 9

⇒ b^{2} = 16 - 9 = 7

Thus, the equation of the ellipse is x^{2}/16 + y^{2}/7 = 1.

**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.**

**Solution**

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 x^{2}/a^{2} + y^{2}/b^{2} = 1, where a is the semi-major axis.

Accordingly, b = 3, c = 4.

It is known that a^{2} = b^{2} + c^{2}.

∴ a^{2} = 3^{2} + 4^{2} = 9 + 16 = 25

⇒ a = 5

Thus, the equation of the ellipse is

**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).**

**Solution**

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

Where, a is the semi - major axis

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 x^{2}/10 + y^{2}/40 = 1 or 4x^{2} + y^{2} = 40.

**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).**

**Solution**

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

Where, a is the semi - major axis

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 x^{2}/52 + y^{2}/13 = 1 or x^{2} + 4y^{2} = 52.