# GSEB Solutions Class 11 Maths Chapter 11 Conic Sections Ex 11.3

Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 11 Conic Sections Ex 11.3 Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 11 Maths Chapter 11 Conic Sections Ex 11.3

In each of the following questions 1 to 9, find the co-ordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and length of latus rectum of the ellipse:
1. $$\frac{x^{2}}{36}$$ + $$\frac{y^{2}}{16}$$ = 1
2. $$\frac{x^{2}}{4}$$ + $$\frac{y^{2}}{25}$$ = 1
3. $$\frac{x^{2}}{16}$$ + $$\frac{y^{2}}{9}$$ = 1
4. $$\frac{x^{2}}{25}$$ + $$\frac{y^{2}}{100}$$ = 1
5. $$\frac{x^{2}}{49}$$ + $$\frac{y^{2}}{36}$$ = 1
6. $$\frac{x^{2}}{100}$$ + $$\frac{y^{2}}{400}$$ = 1
7. 36x2 + 4y2 = 144
8. 16x2 + y2 = 16
9. 4x2 + 9y2 = 36.
Solutions to questions 1 to 9:
1. Equation of ellipse is
$$\frac{x^{2}}{36}$$ + $$\frac{y^{2}}{16}$$ = 1.
Here, a2 = 36, b2 = 16.
∴ a = 6, b = 4
c2 = a2 – b2 = 36 – 16 = 20.
c = ± $$\sqrt{20}$$, = ±2 $$\sqrt{5}$$.
c = ae.
∴ e = $$\frac{c}{a}$$ = $$\frac{2 \sqrt{5}}{6}$$ = $$\frac{\sqrt{5}}{3}$$
Co-ordinates of foci are (± c, 0) i.e., (± 2$$\sqrt{5}$$, 0).
Vertices are (± a, 0) i.e., (± 6, 0).
Length of major axis = 2a = 2 × 6 = 12.
Length of minor axis = 2b = 2 × 4 = 8.
Eccentricity, e = $$\frac{c}{a}$$ = $$\frac{2 \sqrt{5}}{6}$$ = $$\frac{\sqrt{5}}{3}$$
Latus rectum = $$\frac{2 b^{2}}{a}$$ = $$\frac{2×16}{6}$$ = $$\frac{16}{3}$$.

2. Equation of ellipse is $$\frac{x^{2}}{4}$$ + $$\frac{y^{2}}{25}$$ = 1.
Here, b2 = 4 ⇒ b = 2.
and a2 = 25 ⇒ a = 5.
Major axis is along y-axis.
c2 = 25 – 4 = 21
∴ c = $$\sqrt{21}$$.
Co-ordinates of foci are (0, ± c), i.e., (0, ± $$\sqrt{21}$$).
Vertices are (0, ± a) i.e., (0, ± 5).
Length of major axis = 2a = 2 × 5 = 10.
Length of minor axis = 2b = 2 × 2 = 4.
Eccentricity e = $$\frac{c}{a}$$ = $$\frac{\sqrt{21}}{5}$$
Latus rectum = $$\frac{2 b^{2}}{a}$$ = $$\frac{2×4}{5}$$ = $$\frac{8}{5}$$.

3. Equation of ellipse is $$\frac{x^{2}}{16}$$ + $$\frac{y^{2}}{9}$$ = 1.
Here, a2 = 16 ⇒ a = 4 and b2 = 9 ⇒ b = 3.
Major axis is along x-axis.
Also, c2 = a2 – b2 = 16 – 9 = 7 ⇒ c = $$\sqrt{7}$$
Co-ordinates of foci (± c, 0), i.e., (± $$\sqrt{7}$$, 0).
Vertices are (± a, 0), i.e., (± 4, 0).
Length of major axis = 2a = 2 × 4 = 8.
Length of minor axis = 2b = 2 × 3 = 6.
∴ Eccentricity e = $$\frac{c}{a}$$ = $$\frac{\sqrt{7}}{4}$$
Also, Latus rectum = $$\frac{2 b^{2}}{a}$$ = $$\frac{2×9}{4}$$ = $$\frac{9}{2}$$.

4. Equation of ellipse is $$\frac{x^{2}}{25}$$ + $$\frac{y^{2}}{100}$$ = 1.
Major axis is along y-axis.
a2 = 100 ⇒ a = 10, b2 = 25 ⇒ b = 5.
∴ c2 = a2 – b2 = 100 – 25 = 75
∴ c = 5$$\sqrt{3}$$
Foci are (0, ±c), i.e; (0, ± 5$$\sqrt{3}$$).
Vertices are (0, ±c), i.e; (0, ±5$$\sqrt{3}$$).
Length of major axis = 2a = 2 × 10 = 20.
Length of minor axis = 2b = 2 × 5 = 10.
e = $$\frac{c}{a}$$ = $$\frac{5 \sqrt{3}}{10}$$ = $$\frac{\sqrt{3}}{2}$$.
Length of latus rectum = $$\frac{2 b^{2}}{a}$$ = $$\frac{2×25}{10}$$ = 5.

5. $$\frac{x^{2}}{49}$$ + $$\frac{y^{2}}{36}$$ = 1 is the equation of ellipse.
Here, major axis is along x-axis
and a2 = 49 ⇒ a = 7, b2 = 36 ⇒ b = 6.
∴ c2 = a2 – b2 = 49 – 36 = 13.
∴ c = $$\sqrt{13}$$
Foci are (± c, 0), i.e., ($$\sqrt{13}$$, 0).
Vertices are (± a, 0) i.e., (± 7, 0)
Length of major axis = 2a = 2 × 7 = 14.
Length of minor axis = 2b = 2 × 6 = 12.
Length of latus rectum = $$\frac{2 b^{2}}{a}$$ = $$\frac{2×36}{7}$$ = $$\frac{72}{7}$$
Eccentricity, e = $$\frac{c}{a}$$ = $$\frac{\sqrt{13}}{7}$$.

6. $$\frac{x^{2}}{100}$$ + $$\frac{y^{2}}{400}$$ = 1 is the equation of the ellipse.
Major axis is along y-axis.
a2 = 400. ⇒ a = 20, b2 = 100 ⇒ b = 10.
∴ c2 = a2 – b2 = 400 – 100 = 300
∴ c = 10$$\sqrt{3}$$.
Vertices are (0, ± a), i.e., (0, ± 20).
∴ Foci are (0, ± c) i.e., (0, ± 10$$\sqrt{3}$$).
Length of major axis = 2a = 2 × 20 = 40.
Length of minor axis = 2b = 2 × 10 = 20.
Eccentricity = $$\frac{c}{a}$$ = $$\frac{10 \sqrt{3}}{20}$$ = $$\frac{\sqrt{3}}{2}$$.
Length of latus rectum = $$\frac{2 b^{2}}{a}$$ = $$\frac{2×100}{20}$$ = 10.

7. 36x2 + 4y2 = 144 is the equation of the ellipse.
Dividing by 144, we get
$$\frac{x^{2}}{4}$$ + $$\frac{y^{2}}{36}$$ = 1.
Major axis is along y-axis, a2 = 36 or a = 6, b2 = 4 ⇒ b = 2.
∴ c2 = a2 – b2 = 36 – 4 = 32
∴ c = 4$$\sqrt{2}$$.
Foci are (0, ± c), i.e., (0, ± 4$$\sqrt{2}$$).
Vertices are (0, ± a), i.e., (0 ± 6).
Length of major axis = 2a = 2 × 6 = 12.
Length of minor axis = 2b = 2 × 2 = 4.
Eccentricity e = $$\frac{c}{a}$$ = $$\frac{4 \sqrt{2}}{6}$$ = $$\frac{2 \sqrt{2}}{3}$$.
Length of latus rectum = $$\frac{2 b^{2}}{a}$$ = $$\frac{2×4}{6}$$ = $$\frac{4}{3}$$.

8. The equation of the ellipse is 16x2 + y2 = 16.
Dividing by 16, we get
$$\frac{x^{2}}{1}$$ + $$\frac{y^{2}}{16}$$ = 1.
Major axis is along y-axis.
a2 = 16 ⇒ a = 4, b2 = 1 ⇒ b = 1.
and so c2 = a2 – b2 = 16 – 1 = 15.
∴ c = $$\sqrt{15}$$.
Vertices are (0, ± a), i.e., (0, ± 4).
Length of major axis = 2a = 2 × 4 = 8.
Length of minor axis = 2b = 2 × 1 = 2.
Eccentricity e = $$\frac{c}{a}$$ = $$\frac{\sqrt{15}}{4}$$.
Length of latus rectum = $$\frac{2 b^{2}}{a}$$ = $$\frac{2×1}{4}$$ = $$\frac{1}{2}$$.

9. Equation of ellipse is 4x2 + 9y2 = 36.
or $$\frac{x^{2}}{9}$$ + $$\frac{y^{2}}{4}$$ = 1
Major axis is along x-axis.
a2 = 9 ⇒ a = 3, b2 = 4 ⇒ b = 2.
∴ c2 = a2 – b2 = 9 – 4 = 5 ⇒ c = $$\sqrt{5}$$.
Foci are (± c, 0) i.e., (± $$\sqrt{5}$$, 0).
Vertices are (± a, 0), i.e., (± 3, 0).
Length of major axis = 2a = 2 × 3 = 6.
Length of minor axis = 2b = 2 × 2 = 4.
Eccentricity e = $$\frac{c}{a}$$ = $$\frac{\sqrt{5}}{3}$$.
Length of latus rectum = $$\frac{2 b^{2}}{a}$$ = $$\frac{2×4}{3}$$ = $$\frac{8}{3}$$.

In each of the following questions 10 to 20, find the equation for the ellipse that satisfies the given conditions:
10. Vertices (± 5, 0); Foci (± 4, 0)
11. Vertices (0, ± 13); Foci (0, ± 5)
12. Vertices (± 6, 0); Foci (± 4, 0)
13. Ends of major axis (± 3, 0) and ends of minor axis (0, ± 2).
14. Ends of major axis (0, ± $$\sqrt{5}$$) and ends of minor axis (± 1, 0).
15. Length of major axis 26, Foci (± 5, 0).
16. Length of minor axis = 16, Foci (0, ± 6)
17. Foci (± 3, 0); a = 4.
18. b = 3, c = 4, centre at the origin, foci on x-axis.
19. Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).
20. Major axis on x-axis and passes through (4, 3) and (6, 2).
Solutions questions 10-20:
10. Vertices (± 5, 0), Foci (± 4, 0).
⇒ (± a, 0) = (± 5, 0) and (± ae, 0) = (± 4, 0).
∴ a = 5 and ae = 4.
⇒ e = $$\frac{4}{a}$$ = $$\frac{4}{5}$$.
Also, b2 = a2(1 – e2) [Given]
∴ b2 = 25(1 – $$\frac{16}{25}$$) – 16 = 9.
⇒ b = 3.
∴ The equation of ellipse $$\frac{x^{2}}{a^{2}}$$ + $$\frac{y^{2}}{b^{2}}$$ = 1 becomes
$$\frac{x^{2}}{25}$$ + $$\frac{y^{2}}{9}$$ = 1 ⇒ 9x2 + 25y2 = 225.
which is the equation of the required ellipse,

11. Foci (0, ± 5), vertices (0, ± 13)
(0, ± ae) = (0, ± 5) and (0, ± a) = (0, ± 13)
⇒ ae = 5 and a = 13 ∴ e = $$\frac{ae}{a}$$ = $$\frac{5}{13}$$.
b2 = a2 – a2e2 = 132 – 52 = 169 – 25 = 144.
∴ b = 12.
∴ Equation of the required ellipse = $$\frac{x^{2}}{144}$$ + $$\frac{y^{2}}{169}$$ = 1.

12. Vertices and foci of the ellipse are (± 6, 0) and (± 4, 0) respectively.
Major axis is the x-axis.
Vertices are (± 6, 0). Foci are (± 4, 0) ⇒ c = 4 ⇒ a = 6
Now c2 = a2 – b2 or b2 = a2 – c2 = 36 – 16 = 20.
Equation of the ellipse
$$\frac{x^{2}}{36}$$ + $$\frac{y^{2}}{20}$$ = 1.

13. Ends of major axis are (± 3, 0).
⇒ a = 3 and major axis is x-axis.
Ends of minor axis are (0, ± 2).
⇒ b = 2.
∴ Equation of the ellipse is
$$\frac{x^{2}}{9}$$ + $$\frac{y^{2}}{4}$$ = 1.

14. Ends of major axis (0, ± $$\sqrt{5}$$).
Major axis is the y-axis and a2 = $$\sqrt{5}$$.
Ends of minor axis are (± 1, 0)
∴ b = 1.
Equation of ellipse is
$$\frac{x^{2}}{1}$$ + $$\frac{y^{2}}{5}$$ = 1.

15. Length of major axis = 2a = 26.
∴ a = 13
Foci are (± 5, 0), c = 5, ⇒ b2 = a2 – c2.
= 169 – 25
= 144.
Major axis is x-axis.
∴ Equation of ellipse is
$$\frac{x^{2}}{64}$$ + $$\frac{y^{2}}{100}$$ = 1.

16. Length of minor axis = 2b = 16 ⇒ b = 8.
Foci are (0, ± 6) ⇒ c = 6
∴ a2 = b2 + c2
= 64 + 36
= 100.
Major axis is y-axis.
∴ Equation of ellipse
$$\frac{x^{2}}{64}$$ + $$\frac{y^{2}}{100}$$ = 1.

17. Foci are (± 3, 0) ⇒ c = 3.
Also a = 4
∴ b2 = a2 – c2 = 16 – 9 = 7.
Major axis is x-axis and focus lies on it.
Equation of ellipse is
$$\frac{x^{2}}{16}$$ + $$\frac{y^{2}}{7}$$ = 1.

18. b = 3, c = 4 ⇒ a2 = b2 + c2
= 9 + 16 = 25.
Foci are on x-axis.
∴ Major axis is x-axis.
∴ Equation of ellipse is
$$\frac{x^{2}}{25}$$ + $$\frac{y^{2}}{9}$$ = 1.

19. Major axis is y-axis.
Let the ellipse be $$\frac{x^{2}}{b^{2}}$$ + $$\frac{y^{2}}{a^{2}}$$ = 1.
(3, 2) and (1, 6) lies on it.

Subtracting we get,

or 8a2 = 32b2.
∴ a2 = 4b2.
Putting this value in (1), we get
$$\frac{9}{b^{2}}$$ + $$\frac{4}{4b^{2}}$$ = 1 ⇒ $$\frac{10}{b^{2}}$$ = 1
∴ b2 = 10.
Now, a2 = 4b2 = 4 × 10 = 40.
∴ Equation of the ellipse is
$$\frac{x^{2}}{10}$$ + $$\frac{y^{2}}{40}$$ = 1.

20. Major axis is x-axis.
Let the equation of the ellipse be
$$\frac{x^{2}}{a^{2}}$$ + $$\frac{y^{2}}{b^{2}}$$ = 1.
(4, 3) and (6, 2) lies on it.
Therefore,

Subtracting (2) from (1), we get
$$\frac{- 20}{a^{2}}$$ + $$\frac{5}{b^{2}}$$ = 0
or 5a2 = 20b2 or a2 = 4b2.
Putting a2 = 5b2 in (1), we get
$$\frac{16}{4b^{2}}$$ + $$\frac{9}{b^{2}}$$ = 1 ⇒ b2 = 13
and a2 = 4b2 = 4 × 13 = 52.
∴ Equation of the ellipse is
$$\frac{x^{2}}{52}$$ + $$\frac{y^{2}}{13}$$ = 1.