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

8. 16x^{2} + y^{2} = 16

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

Solutions to questions 1 to 9:

1. Equation of ellipse is

\(\frac{x^{2}}{36}\) + \(\frac{y^{2}}{16}\) = 1.

Here, a^{2} = 36, b^{2} = 16.

∴ a = 6, b = 4

c^{2} = a^{2} – b^{2} = 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, b^{2} = 4 ⇒ b = 2.

and a^{2} = 25 ⇒ a = 5.

Major axis is along y-axis.

c^{2} = 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, a^{2} = 16 ⇒ a = 4 and b^{2} = 9 ⇒ b = 3.

Major axis is along x-axis.

Also, c^{2} = a^{2} – b^{2} = 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.

a^{2} = 100 ⇒ a = 10, b^{2} = 25 ⇒ b = 5.

∴ c^{2} = a^{2} – b^{2} = 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 a^{2} = 49 ⇒ a = 7, b^{2} = 36 ⇒ b = 6.

∴ c^{2} = a^{2} – b^{2} = 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.

a^{2} = 400. ⇒ a = 20, b^{2} = 100 ⇒ b = 10.

∴ c^{2} = a^{2} – b^{2} = 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. 36x^{2} + 4y^{2} = 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, a^{2} = 36 or a = 6, b^{2} = 4 ⇒ b = 2.

∴ c^{2} = a^{2} – b^{2} = 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 16x^{2} + y^{2} = 16.

Dividing by 16, we get

\(\frac{x^{2}}{1}\) + \(\frac{y^{2}}{16}\) = 1.

Major axis is along y-axis.

a^{2} = 16 ⇒ a = 4, b^{2} = 1 ⇒ b = 1.

and so c^{2} = a^{2} – b^{2} = 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 4x^{2} + 9y^{2} = 36.

or \(\frac{x^{2}}{9}\) + \(\frac{y^{2}}{4}\) = 1

Major axis is along x-axis.

a^{2} = 9 ⇒ a = 3, b^{2} = 4 ⇒ b = 2.

∴ c^{2} = a^{2} – b^{2} = 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, b^{2} = a^{2}(1 – e^{2}) [Given]

∴ b^{2} = 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 ⇒ 9x^{2} + 25y^{2} = 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 = \(\frac{ae}{a}\) = \(\frac{5}{13}\).

b^{2} = a^{2} – a^{2}e^{2} = 13^{2} – 5^{2} = 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 c^{2} = a^{2} – b^{2} or b^{2} = a^{2} – c^{2} = 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 a^{2} = \(\sqrt{5}\).

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, ⇒ b^{2} = a^{2} – c^{2}.

= 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

∴ a^{2} = b^{2} + c^{2}

= 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

∴ b^{2} = a^{2} – c^{2} = 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 ⇒ a^{2} = b^{2} + c^{2}

= 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 8a^{2} = 32b^{2}.

∴ a^{2} = 4b^{2}.

Putting this value in (1), we get

\(\frac{9}{b^{2}}\) + \(\frac{4}{4b^{2}}\) = 1 ⇒ \(\frac{10}{b^{2}}\) = 1

∴ b^{2} = 10.

Now, a^{2} = 4b^{2} = 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 5a^{2} = 20b^{2} or a^{2} = 4b^{2}.

Putting a^{2} = 5b^{2} in (1), we get

\(\frac{16}{4b^{2}}\) + \(\frac{9}{b^{2}}\) = 1 ⇒ b^{2} = 13

and a^{2} = 4b^{2} = 4 × 13 = 52.

∴ Equation of the ellipse is

\(\frac{x^{2}}{52}\) + \(\frac{y^{2}}{13}\) = 1.