Question

Let the centre of an ellipse is at $(0,0)$
Assertion: If major axis is on the y-axis and ellipse passes through the points $(3,2)$ and $(1,6)$, then the equation of ellipse is $\frac{x^{2}}{10}+\frac{y^{2}}{40}=1$.
Reason: $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$ is an equation of ellipse if major axis is along y-axis.

• A. Assertion is correct, reason is correct; reason is a correct explanation for assertion.
• B. Assertion is correct, reason is correct; reason is not a correct explanation for assertion
• C. Assertion is correct, reason is incorrect
• D. Assertion is incorrect, reason is correct.

Answer 1

Answer: (A)

Assertion: Since, major axis is along y-axis. Hence, equation of ellipse will be of the form $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1 \,\,\,\,\,\,\,\,\, ....(i)$
Given that (i) passes through the points $(3,2)$ and $(1,6)$ i.e, they will satisfy it
$\therefore \frac{3^{2}}{ b ^{2}}+\frac{2^{2}}{ a ^{2}}=1 \Rightarrow \frac{9}{ b ^{2}}+\frac{4}{ a ^{2}}=1 \,\,\,\,\,\,\,\,\, ....(ii) $
and $\frac{1^{2}}{b^{2}}+\frac{6^{2}}{a^{2}}=1 \Rightarrow \frac{1}{b^{2}}+\frac{36}{a^{2}}=1 \,\,\,\,\,\,\,\,\, ....(iii)$
Multiplying (ii) by 9 and then subtracting (iii) from it,
we get $\frac{80}{ b ^{2}}=8 \Rightarrow b ^{2}=\frac{80}{8} \therefore b ^{2}=10$
from eq (ii), we get $\frac{9}{10}+\frac{4}{ a ^{2}}=1 \Rightarrow \frac{4}{ a ^{2}}=1-\frac{9}{10} \Rightarrow a ^{2}=40 $
putting the value of $a ^{2}=40$ and $b ^{2}=10$ in (i), we get $\frac{x^{2}}{10}+\frac{y^{2}}{40}=1$