Ellipse
Ellipse from Latus Rectum and Angle Conditions
nta_pyq_2023_apr
Grade 11
Question:
Let an ellipse with centre $(1,0)$ and latus rectum of length $\dfrac{1}{2}$ have its major axis along the $x$-axis. If its minor axis subtends an angle $60°$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to ___.
Step-by-Step Solution
Key Concept: LR $=\frac{2b^2}{a}=\frac{1}{2}\Rightarrow b^2=\frac{a}{4}$. Minor axis subtends $60°$ at each focus: $\tan30°=\frac{b}{ae}$, so $b^2=\frac{a^2-b^2}{3}\Rightarrow b^2=\frac{a^2}{4}$.
$a=1,\ b=\frac{1}{2}$. $(2a+2b)^2=9$.
Correct Answer: 9