Anand Classes provides NCERT Solutions for Class 11 Maths Chapter 10 – Conic Sections (Exercise 10.3: Ellipse) following the latest NCERT and CBSE syllabus (2025–2026). This exercise explains the definition and standard equation of an ellipse, along with important terms such as foci, major axis, minor axis, eccentricity, and center. Students will also learn how to derive and solve problems on equations of ellipses in standard and general form. Each solution is presented step-by-step to help strengthen concepts in coordinate geometry and prepare for CBSE board exams, JEE Main, JEE Advanced, NDA, and CUET. Click the print button to download study material and notes.
NCERT Question 1 : Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $$\frac{x^2}{36}+\frac{y^2}{16}=1$$
Solution :
Given equation of ellipse:
$$\frac{x^2}{36}+\frac{y^2}{16}=1$$
Since denominator of $x^2/36$ is larger than that of $y^2/16$, the major axis is along the $x$-axis.
Compare with $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$.
$a^2=36,; b^2=16\Rightarrow a=6,; b=4$.
$c=\sqrt{a^2-b^2}=\sqrt{36-16}=\sqrt{20}=2\sqrt{5}$.
Foci: $(\pm c,0)=(\pm 2\sqrt{5},0)$.
Vertices: $(\pm a,0)=(\pm 6,0)$.
Length of major axis $=2a=12$.
Length of minor axis $=2b=8$.
Eccentricity $e=\dfrac{c}{a}=\dfrac{2\sqrt{5}}{6}=\dfrac{\sqrt{5}}{3}$.
Latus rectum $=\dfrac{2b^2}{a}=\dfrac{2\cdot 16}{6}=\dfrac{16}{3}$.
NCERT Question 2 : Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse
$$\frac{x^2}{4}+\frac{y^2}{25}=1$$
Solution
Denominator of $y^2/25$ is larger ⇒ major axis along the $y$-axis.
$a^2=4,; b^2=25\Rightarrow a=2,; b=5$ (here $b>a$ so use $b$ for major).
$c=\sqrt{b^2-a^2}=\sqrt{25-4}=\sqrt{21}$.
Foci: $(0,\pm c)=(0,\pm\sqrt{21})$.
Vertices: $(0,\pm b)=(0,\pm 5)$.
Length of major axis $=2b=10$.
Length of minor axis $=2a=4$.
Eccentricity $e=\dfrac{c}{b}=\dfrac{\sqrt{21}}{5}$.
Latus rectum $=\dfrac{2a^2}{b}=\dfrac{2\cdot 4}{5}=\dfrac{8}{5}$.
NCERT Question 3 : Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse
$$\frac{x^2}{16}+\frac{y^2}{9}=1$$
Solution :
Major axis along $x$ (since $16>9$).
$a^2=16,; b^2=9\Rightarrow a=4,; b=3$.
$c=\sqrt{16-9}=\sqrt{7}$.
Foci: $(\pm\sqrt{7},0)$.
Vertices: $(\pm 4,0)$.
Length of major axis $=2a=8$.
Length of minor axis $=2b=6$.
Eccentricity $e=\dfrac{\sqrt{7}}{4}$.
Latus rectum $=\dfrac{2b^2}{a}=\dfrac{2\cdot 9}{4}=\dfrac{9}{2}$.
NCERT Question 4 : Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse
$$\frac{x^2}{25}+\frac{y^2}{100}=1$$
Solution :
Major axis along $y$ (since $100>25$).
$a^2=25,; b^2=100\Rightarrow a=5,; b=10$ (here $b$ is major).
$c=\sqrt{b^2-a^2}=\sqrt{100-25}=\sqrt{75}=5\sqrt{3}$.
Foci: $(0,\pm 5\sqrt{3})$.
Vertices: $(0,\pm 10)$.
Length of major axis $=2b=20$.
Length of minor axis $=2a=10$.
Eccentricity $e=\dfrac{c}{b}=\dfrac{5\sqrt{3}}{10}=\dfrac{\sqrt{3}}{2}$.
Latus rectum $=\dfrac{2a^2}{b}=\dfrac{2\cdot 25}{10}=5$.
NCERT Question 5 : Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse
$$\frac{x^2}{49}+\frac{y^2}{36}=1$$
Solution :
Major axis along $x$ ($49>36$).
$a^2=49,; b^2=36\Rightarrow a=7,; b=6$.
$c=\sqrt{49-36}=\sqrt{13}$.
Foci: $(\pm\sqrt{13},0)$.
Vertices: $(\pm 7,0)$.
Length of major axis $=2a=14$.
Length of minor axis $=2b=12$.
Eccentricity $e=\dfrac{\sqrt{13}}{7}$.
Latus rectum $=\dfrac{2b^2}{a}=\dfrac{2\cdot 36}{7}=\dfrac{72}{7}$.
NCERT Question 6 : Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse
$$\frac{x^2}{100}+\frac{y^2}{400}=1$$
Solution :
Major axis along $y$ ($400>100$).
$a^2=100,; b^2=400\Rightarrow a=10,; b=20$ (here $b$ major).
$c=\sqrt{400-100}=\sqrt{300}=10\sqrt{3}$.
Foci: $(0,\pm 10\sqrt{3})$.
Vertices: $(0,\pm 20)$.
Length of major axis $=2b=40$.
Length of minor axis $=2a=20$.
Eccentricity $e=\dfrac{c}{b}=\dfrac{10\sqrt{3}}{20}=\dfrac{\sqrt{3}}{2}$.
Latus rectum $=\dfrac{2a^2}{b}=\dfrac{2\cdot 100}{20}=10$.
NCERT Question 7 : Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $$36x^2+4y^2=144$$
Solution :
Equation given: $36x^2+4y^2=144$
Divide both sides by $144$ to get standard form:
$$\frac{x^2}{4}+\frac{y^2}{36}=1$$
Major axis along $y$ ($36>4$).
$a^2=4,; b^2=36\Rightarrow a=2,; b=6$ (here $b$ major).
$c=\sqrt{36-4}=\sqrt{32}=4\sqrt{2}$.
Foci: $(0,\pm 4\sqrt{2})$.
Vertices: $(0,\pm 6)$.
Length of major axis $=2b=12$.
Length of minor axis $=2a=4$.
Eccentricity $e=\dfrac{c}{b}=\dfrac{4\sqrt{2}}{6}=\dfrac{2\sqrt{2}}{3}$.
Latus rectum $=\dfrac{2a^2}{b}=\dfrac{2\cdot 4}{6}=\dfrac{4}{3}$.
NCERT Question 8 : Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $$16x^2+y^2=16$$
Solution :
Equation given: $16x^2+y^2=16$
Divide by $16$:
$$\frac{x^2}{1}+\frac{y^2}{16}=1$$
Major axis along $y$ ($16>1$).
$a^2=1,; b^2=16\Rightarrow a=1,; b=4$ (here $b$ major).
$c=\sqrt{16-1}=\sqrt{15}$.
Foci: $(0,\pm\sqrt{15})$.
Vertices: $(0,\pm 4)$.
Length of major axis $=2b=8$.
Length of minor axis $=2a=2$.
Eccentricity $e=\dfrac{\sqrt{15}}{4}$.
Latus rectum $=\dfrac{2a^2}{b}=\dfrac{2\cdot 1}{4}=\dfrac{1}{2}$.
NCERT Question 9 : Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $$4x^2+9y^2=36$$
Solution :
Equation given: $4x^2+9y^2=36$
Divide by $36$:
$$\frac{x^2}{9}+\frac{y^2}{4}=1$$
Major axis along $x$ ($9>4$).
$a^2=9,; b^2=4\Rightarrow a=3,; b=2$.
$c=\sqrt{9-4}=\sqrt{5}$.
Foci: $(\pm\sqrt{5},0)$.
Vertices: $(\pm 3,0)$.
Length of major axis $=2a=6$.
Length of minor axis $=2b=4$.
Eccentricity $e=\dfrac{\sqrt{5}}{3}$.
Latus rectum $=\dfrac{2b^2}{a}=\dfrac{2\cdot 4}{3}=\dfrac{8}{3}$.
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