Class 11 Maths Chapter 10 Conic Sections – Exercise 10.3 NCERT Solutions

Introduction

Exercise 10.3 introduces the ellipse as a conic section. You will learn its definition, standard equations, and properties. Problems focus on finding equations of ellipses, identifying foci, vertices, and eccentricity, and solving coordinate geometry questions involving ellipses.

Key Concepts

1. Definition of Ellipse: Locus of a point such that the sum of its distances from two fixed points (foci) is constant.

2. Standard Equations of Ellipse:

   • Major axis along x‑axis:

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$$

• Major axis along y‑axis:

$$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1,a>b$$

3. Key Elements:

   • Vertices: $(\pm a,0)$ or $(0,\pm a)$.

   • Foci: $(\pm c,0)$ or $(0,\pm c)$, where $c^2=a^2-b^2$.

   • Eccentricity: $e=\frac{c}{a}$.

NCERT Questions with Solutions (10)

Q1. Find equation of ellipse with foci on x‑axis, center at origin, $a=5,b=3$.

$$\frac{x^2}{25}+\frac{y^2}{9}=1$$

Q2. Find equation of ellipse with foci on y‑axis, center at origin, $a=6,b=4$.

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

Q3. Find vertices and foci of ellipse $\frac{x^2}{49}+\frac{y^2}{25}=1$. Vertices: $(\pm 7,0)$. Foci: $(\pm \sqrt{49-25},0)=(\pm \sqrt{24},0)=(\pm 2\sqrt{6},0)$.

Q4. Find vertices and foci of ellipse $\frac{x^2}{16}+\frac{y^2}{36}=1$. Vertices: $(0,\pm 6)$. Foci: $(0,\pm \sqrt{36-16})=(0,\pm \sqrt{20})=(0,\pm 2\sqrt{5})$.

Q5. Find eccentricity of ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$.

$$c^2=a^2-b^2=25-16=9⟹c=3$$

$$e=\frac{c}{a}=\frac{3}{5}$$

Q6. Find eccentricity of ellipse $\frac{x^2}{36}+\frac{y^2}{9}=1$.

$$c^2=36-9=27⟹c=3\sqrt{3}$$

$$e=\frac{c}{a}=\frac{3\sqrt{3}}{6}=\frac{\sqrt{3}}{2}$$

Q7. Find equation of ellipse with vertices (±10,0) and minor axis length 12.

$$a=10,b=6$$

Equation:

$$\frac{x^2}{100}+\frac{y^2}{36}=1$$

Q8. Find equation of ellipse with vertices (0,±8) and minor axis length 6.

$$a=8,b=3$$

Equation:

$$\frac{x^2}{9}+\frac{y^2}{64}=1$$

Q9. Find equation of ellipse with foci (±4,0) and major axis length 10.

$$a=5,c=4⟹b^2=a^2-c^2=25-16=9$$

Equation:

$$\frac{x^2}{25}+\frac{y^2}{9}=1$$

Q10. Find equation of ellipse with foci (0,±3) and major axis length 8.

$$a=4,c=3⟹b^2=a^2-c^2=16-9=7$$

Equation:

$$\frac{x^2}{7}+\frac{y^2}{16}=1$$

FAQs (10)

1. What is ellipse? Locus of point with constant sum of distances from two foci.

2. What is standard equation of ellipse? $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ or $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$.

3. What are vertices of ellipse? Endpoints of major axis.

4. What are foci of ellipse? Fixed points inside ellipse.

5. What is eccentricity? Ratio $e=\frac{c}{a}$.

6. What is relation between $a,b,c$? $c^2=a^2-b^2$.

7. Can ellipse be circle? Yes, if $a=b$.

8. What is major axis? Longer axis of ellipse.

9. What is minor axis? Shorter axis of ellipse.

10. Why is this exercise important? It builds foundation for ellipse properties and applications in geometry and physics.

Conclusion

Exercise 10.3 has 10 solved questions and 10 FAQs that strengthen your understanding of ellipses in coordinate geometry. This builds the foundation for hyperbola and other conic sections in Class 11 Maths.

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New Syllabus-Class 11 Maths Chapter 10 Conic Sections – Exercise 10.3 NCERT Solutions