Rbse Solutions for Class 11 maths Chapter 10 Exercise 10.3 | Ellipse

Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 10 Exercise 10.3

This exercise covers finding the key properties of ellipses from their equations and determining the equation of an ellipse from given properties. All ellipses here are centered at the origin $(0, 0)$.

The key relation for any ellipse is $a^2 = b^2 + c^2$, where $a$ is the semi-major axis, $b$ is the semi-minor axis, and $c$ is the distance from the center to the focus. Eccentricity is $e = c/a$.

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Finding Properties from the Equation (Exercises 1–9)

The two standard forms are:

1. Horizontal Major Axis: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a^2 > b^2$)
2. Vertical Major Axis: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ (where $a^2 > b^2$)

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

• Form: Horizontal (since $36 > 16$, $a^2$ is under $x^2$).
• $a^2 = 36 \implies \mathbf{a = 6}$.
• $b^2 = 16 \implies \mathbf{b = 4}$.
• Foci: $c^2 = a^2 – b^2 = 36 – 16 = 20 \implies c = \sqrt{20} = 2\sqrt{5}$. Foci are $(\pm c, 0) = \mathbf{(\pm 2\sqrt{5}, 0)}$.
• Vertices: $(\pm a, 0) = \mathbf{(\pm 6, 0)}$.
• Major Axis Length: $2a = \mathbf{12}$.
• Minor Axis Length: $2b = \mathbf{8}$.
• Eccentricity: $e = c/a = \mathbf{\frac{2\sqrt{5}}{6} = \frac{\sqrt{5}}{3}}$.
• Latus Rectum Length: $\frac{2b^2}{a} = \frac{2(16)}{6} = \mathbf{\frac{16}{3}}$.

2. $\frac{x^2}{4} + \frac{y^2}{25} = 1$

• Form: Vertical (since $25 > 4$, $a^2$ is under $y^2$).
• $a^2 = 25 \implies \mathbf{a = 5}$.
• $b^2 = 4 \implies \mathbf{b = 2}$.
• Foci: $c^2 = a^2 – b^2 = 25 – 4 = 21 \implies c = \sqrt{21}$. Foci are $(0, \pm c) = \mathbf{(0, \pm \sqrt{21})}$.
• Vertices: $(0, \pm a) = \mathbf{(0, \pm 5)}$.
• Major Axis Length: $2a = \mathbf{10}$.
• Minor Axis Length: $2b = \mathbf{4}$.
• Eccentricity: $e = c/a = \mathbf{\frac{\sqrt{21}}{5}}$.
• Latus Rectum Length: $\frac{2b^2}{a} = \frac{2(4)}{5} = \mathbf{\frac{8}{5}}$.

3. $\frac{x^2}{16} + \frac{y^2}{9} = 1$

• Form: Horizontal (since $16 > 9$).
• $a^2 = 16 \implies \mathbf{a = 4}$.
• $b^2 = 9 \implies \mathbf{b = 3}$.
• Foci: $c^2 = 16 – 9 = 7 \implies c = \sqrt{7}$. Foci are $(\pm c, 0) = \mathbf{(\pm \sqrt{7}, 0)}$.
• Vertices: $(\pm a, 0) = \mathbf{(\pm 4, 0)}$.
• Major Axis Length: $2a = \mathbf{8}$.
• Minor Axis Length: $2b = \mathbf{6}$.
• Eccentricity: $e = c/a = \mathbf{\frac{\sqrt{7}}{4}}$.
• Latus Rectum Length: $\frac{2b^2}{a} = \frac{2(9)}{4} = \mathbf{\frac{9}{2}}$.

4. $\frac{x^2}{25} + \frac{y^2}{100} = 1$

• Form: Vertical (since $100 > 25$).
• $a^2 = 100 \implies \mathbf{a = 10}$.
• $b^2 = 25 \implies \mathbf{b = 5}$.
• Foci: $c^2 = 100 – 25 = 75 \implies c = \sqrt{75} = 5\sqrt{3}$. Foci are $(0, \pm c) = \mathbf{(0, \pm 5\sqrt{3})}$.
• Vertices: $(0, \pm a) = \mathbf{(0, \pm 10)}$.
• Major Axis Length: $2a = \mathbf{20}$.
• Minor Axis Length: $2b = \mathbf{10}$.
• Eccentricity: $e = c/a = \mathbf{\frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2}}$.
• Latus Rectum Length: $\frac{2b^2}{a} = \frac{2(25)}{10} = \mathbf{5}$.

5. $\frac{x^2}{49} + \frac{y^2}{36} = 1$

• Form: Horizontal (since $49 > 36$).
• $a^2 = 49 \implies \mathbf{a = 7}$.
• $b^2 = 36 \implies \mathbf{b = 6}$.
• Foci: $c^2 = 49 – 36 = 13 \implies c = \sqrt{13}$. Foci are $(\pm c, 0) = \mathbf{(\pm \sqrt{13}, 0)}$.
• Vertices: $(\pm a, 0) = \mathbf{(\pm 7, 0)}$.
• Major Axis Length: $2a = \mathbf{14}$.
• Minor Axis Length: $2b = \mathbf{12}$.
• Eccentricity: $e = c/a = \mathbf{\frac{\sqrt{13}}{7}}$.
• Latus Rectum Length: $\frac{2b^2}{a} = \frac{2(36)}{7} = \mathbf{\frac{72}{7}}$.

6. $\frac{x^2}{100} + \frac{y^2}{400} = 1$

• Form: Vertical (since $400 > 100$).
• $a^2 = 400 \implies \mathbf{a = 20}$.
• $b^2 = 100 \implies \mathbf{b = 10}$.
• Foci: $c^2 = 400 – 100 = 300 \implies c = \sqrt{300} = 10\sqrt{3}$. Foci are $(0, \pm c) = \mathbf{(0, \pm 10\sqrt{3})}$.
• Vertices: $(0, \pm a) = \mathbf{(0, \pm 20)}$.
• Major Axis Length: $2a = \mathbf{40}$.
• Minor Axis Length: $2b = \mathbf{20}$.
• Eccentricity: $e = c/a = \mathbf{\frac{10\sqrt{3}}{20} = \frac{\sqrt{3}}{2}}$.
• Latus Rectum Length: $\frac{2b^2}{a} = \frac{2(100)}{20} = \mathbf{10}$.

7. $36x^2 + 4y^2 = 144$

Divide by 144 to get the standard form:

$$\frac{36x^2}{144} + \frac{4y^2}{144} = 1 \implies \frac{x^2}{4} + \frac{y^2}{36} = 1$$

• Form: Vertical (since $36 > 4$).
• $a^2 = 36 \implies \mathbf{a = 6}$.
• $b^2 = 4 \implies \mathbf{b = 2}$.
• Foci: $c^2 = 36 – 4 = 32 \implies c = \sqrt{32} = 4\sqrt{2}$. Foci are $(0, \pm c) = \mathbf{(0, \pm 4\sqrt{2})}$.
• Vertices: $(0, \pm a) = \mathbf{(0, \pm 6)}$.
• Major Axis Length: $2a = \mathbf{12}$.
• Minor Axis Length: $2b = \mathbf{4}$.
• Eccentricity: $e = c/a = \mathbf{\frac{4\sqrt{2}}{6} = \frac{2\sqrt{2}}{3}}$.
• Latus Rectum Length: $\frac{2b^2}{a} = \frac{2(4)}{6} = \mathbf{\frac{4}{3}}$.

8. $16x^2 + y^2 = 16$

Divide by 16:

$$\frac{16x^2}{16} + \frac{y^2}{16} = 1 \implies \frac{x^2}{1} + \frac{y^2}{16} = 1$$

• Form: Vertical (since $16 > 1$).
• $a^2 = 16 \implies \mathbf{a = 4}$.
• $b^2 = 1 \implies \mathbf{b = 1}$.
• Foci: $c^2 = 16 – 1 = 15 \implies c = \sqrt{15}$. Foci are $(0, \pm c) = \mathbf{(0, \pm \sqrt{15})}$.
• Vertices: $(0, \pm a) = \mathbf{(0, \pm 4)}$.
• Major Axis Length: $2a = \mathbf{8}$.
• Minor Axis Length: $2b = \mathbf{2}$.
• Eccentricity: $e = c/a = \mathbf{\frac{\sqrt{15}}{4}}$.
• Latus Rectum Length: $\frac{2b^2}{a} = \frac{2(1)}{4} = \mathbf{\frac{1}{2}}$.

9. $4x^2 + 9y^2 = 36$

Divide by 36:

$$\frac{4x^2}{36} + \frac{9y^2}{36} = 1 \implies \frac{x^2}{9} + \frac{y^2}{4} = 1$$

• Form: Horizontal (since $9 > 4$).
• $a^2 = 9 \implies \mathbf{a = 3}$.
• $b^2 = 4 \implies \mathbf{b = 2}$.
• Foci: $c^2 = 9 – 4 = 5 \implies c = \sqrt{5}$. Foci are $(\pm c, 0) = \mathbf{(\pm \sqrt{5}, 0)}$.
• Vertices: $(\pm a, 0) = \mathbf{(\pm 3, 0)}$.
• Major Axis Length: $2a = \mathbf{6}$.
• Minor Axis Length: $2b = \mathbf{4}$.
• Eccentricity: $e = c/a = \mathbf{\frac{\sqrt{5}}{3}}$.
• Latus Rectum Length: $\frac{2b^2}{a} = \frac{2(4)}{3} = \mathbf{\frac{8}{3}}$.

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Finding the Equation of the Ellipse (Exercises 10–20)

The equation is $\frac{x^2}{A} + \frac{y^2}{B} = 1$, where $A$ and $B$ are $a^2$ and $b^2$ in some order.

10. Vertices $(\pm 5, 0)$, foci $(\pm 4, 0)$.

1. Axis: Vertices are on the $x$-axis, so the major axis is horizontal. Form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
2. $a$ and $c$: Vertices $(\pm a, 0) \implies \mathbf{a = 5}$. Foci $(\pm c, 0) \implies \mathbf{c = 4}$.
3. $b^2$: $b^2 = a^2 – c^2 = 5^2 – 4^2 = 25 – 16 = \mathbf{9}$.$$\mathbf{\frac{x^2}{25} + \frac{y^2}{9} = 1}$$

11. Vertices $(0, \pm 13)$, foci $(0, \pm 5)$.

1. Axis: Vertices are on the $y$-axis, so the major axis is vertical. Form: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$.
2. $a$ and $c$: Vertices $(0, \pm a) \implies \mathbf{a = 13}$. Foci $(0, \pm c) \implies \mathbf{c = 5}$.
3. $b^2$: $b^2 = a^2 – c^2 = 13^2 – 5^2 = 169 – 25 = \mathbf{144}$.$$\mathbf{\frac{x^2}{144} + \frac{y^2}{169} = 1}$$

12. Vertices $(\pm 6, 0)$, foci $(\pm 4, 0)$.

1. Axis: Horizontal. Form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
2. $a$ and $c$: $\mathbf{a = 6}$, $\mathbf{c = 4}$.
3. $b^2$: $b^2 = a^2 – c^2 = 6^2 – 4^2 = 36 – 16 = \mathbf{20}$.$$\mathbf{\frac{x^2}{36} + \frac{y^2}{20} = 1}$$

13. Ends of major axis $(\pm 3, 0)$, ends of minor axis $(0, \pm 2)$.

1. Axis: Major axis ends are on the $x$-axis, so the major axis is horizontal. Form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
2. $a$ and $b$: Major axis ends $(\pm a, 0) \implies \mathbf{a = 3}$. Minor axis ends $(0, \pm b) \implies \mathbf{b = 2}$.
3. $a^2$ and $b^2$: $a^2 = \mathbf{9}$, $b^2 = \mathbf{4}$.$$\mathbf{\frac{x^2}{9} + \frac{y^2}{4} = 1}$$

14. Ends of major axis $(0, \pm 5)$, ends of minor axis $(\pm 1, 0)$.

1. Axis: Major axis ends are on the $y$-axis, so the major axis is vertical. Form: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$.
2. $a$ and $b$: Major axis ends $(0, \pm a) \implies \mathbf{a = 5}$. Minor axis ends $(\pm b, 0) \implies \mathbf{b = 1}$.
3. $a^2$ and $b^2$: $a^2 = \mathbf{25}$, $b^2 = \mathbf{1}$.$$\mathbf{\frac{x^2}{1} + \frac{y^2}{25} = 1 \quad \text{or} \quad x^2 + \frac{y^2}{25} = 1}$$

15. Length of major axis 26, foci $(\pm 5, 0)$.

1. Axis: Foci are on the $x$-axis, so the major axis is horizontal. Form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
2. $a$ and $c$: Length of major axis $2a = 26 \implies \mathbf{a = 13}$. Foci $(\pm c, 0) \implies \mathbf{c = 5}$.
3. $b^2$: $b^2 = a^2 – c^2 = 13^2 – 5^2 = 169 – 25 = \mathbf{144}$.$$\mathbf{\frac{x^2}{169} + \frac{y^2}{144} = 1}$$

16. Length of minor axis 16, foci $(0, \pm 6)$.

1. Axis: Foci are on the $y$-axis, so the major axis is vertical. Form: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$.
2. $b$ and $c$: Length of minor axis $2b = 16 \implies \mathbf{b = 8}$. Foci $(0, \pm c) \implies \mathbf{c = 6}$.
3. $a^2$: $a^2 = b^2 + c^2 = 8^2 + 6^2 = 64 + 36 = \mathbf{100}$.$$\mathbf{\frac{x^2}{64} + \frac{y^2}{100} = 1}$$

17. Foci $(\pm 3, 0)$, $a = 4$.

1. Axis: Foci are on the $x$-axis, so the major axis is horizontal. Form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
2. $a$ and $c$: $\mathbf{a = 4}$. Foci $(\pm c, 0) \implies \mathbf{c = 3}$.
3. $b^2$: $b^2 = a^2 – c^2 = 4^2 – 3^2 = 16 – 9 = \mathbf{7}$.$$\mathbf{\frac{x^2}{16} + \frac{y^2}{7} = 1}$$

18. $b = 3$, $c = 4$, center at the origin; foci on the $x$-axis.

1. Axis: Foci on the $x$-axis, so the major axis is horizontal. Form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
2. $a^2$: $a^2 = b^2 + c^2 = 3^2 + 4^2 = 9 + 16 = \mathbf{25}$.
3. $b^2$: $b^2 = \mathbf{9}$.$$\mathbf{\frac{x^2}{25} + \frac{y^2}{9} = 1}$$

19. Center at $(0, 0)$, major axis on the $y$-axis and passes through the points $(3, 2)$ and $(1, 6)$.

1. Form: Major axis on the $y$-axis means the ellipse is vertical. Form: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$.
2. Substitute points:
   • Point $(3, 2)$: $\frac{3^2}{b^2} + \frac{2^2}{a^2} = 1 \implies \mathbf{\frac{9}{b^2} + \frac{4}{a^2} = 1} \quad \text{(i)}$
   • Point $(1, 6)$: $\frac{1^2}{b^2} + \frac{6^2}{a^2} = 1 \implies \mathbf{\frac{1}{b^2} + \frac{36}{a^2} = 1} \quad \text{(ii)}$
3. Solve the system: Let $u = 1/a^2$ and $v = 1/b^2$.
   • (i) $4u + 9v = 1$
   • (ii) $36u + 1v = 1$
4. Multiply (ii) by 9: $324u + 9v = 9$. Subtract (i) from this:$$(324u + 9v) – (4u + 9v) = 9 – 1$$$$320u = 8 \implies u = \frac{8}{320} = \mathbf{\frac{1}{40}}$$
5. Find $v$: Substitute $u$ into (ii):$$36\left(\frac{1}{40}\right) + v = 1$$$$v = 1 – \frac{36}{40} = 1 – \frac{9}{10} = \mathbf{\frac{1}{10}}$$
6. Final Equation: $u = 1/a^2 = 1/40 \implies a^2 = 40$. $v = 1/b^2 = 1/10 \implies b^2 = 10$.$$\mathbf{\frac{x^2}{10} + \frac{y^2}{40} = 1}$$

20. Major axis on the $x$-axis and passes through the points $(4, 3)$ and $(6, 2)$.

1. Form: Major axis on the $x$-axis means the ellipse is horizontal. Form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
2. Substitute points:
   • Point $(4, 3)$: $\frac{4^2}{a^2} + \frac{3^2}{b^2} = 1 \implies \mathbf{\frac{16}{a^2} + \frac{9}{b^2} = 1} \quad \text{(i)}$
   • Point $(6, 2)$: $\frac{6^2}{a^2} + \frac{2^2}{b^2} = 1 \implies \mathbf{\frac{36}{a^2} + \frac{4}{b^2} = 1} \quad \text{(ii)}$
3. Solve the system: Let $u = 1/a^2$ and $v = 1/b^2$.
   • (i) $16u + 9v = 1$
   • (ii) $36u + 4v = 1$
4. Multiply (i) by 4 and (ii) by 9 to eliminate $v$:
   • (i’) $64u + 36v = 4$
   • (ii’) $324u + 36v = 9$
5. Subtract (i’) from (ii’):$$(324u + 36v) – (64u + 36v) = 9 – 4$$$$260u = 5 \implies u = \frac{5}{260} = \mathbf{\frac{1}{52}}$$
6. Find $v$: Substitute $u$ into (ii):$$36\left(\frac{1}{52}\right) + 4v = 1$$$$4v = 1 – \frac{36}{52} = 1 – \frac{9}{13} = \frac{4}{13}$$$$v = \frac{1}{13}$$
7. Final Equation: $u = 1/a^2 = 1/52 \implies a^2 = 52$. $v = 1/b^2 = 1/13 \implies b^2 = 13$.$$\mathbf{\frac{x^2}{52} + \frac{y^2}{13} = 1}$$

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