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.