Ellipse
Eccentricity and Latus Rectum from Composite Functions
nta_pyq_2024_apr
Grade 11
Question:
Let $f(x)=x^2+9$, $g(x)=\dfrac{x}{x-9}$ and $a=f\circ g(10)$, $b=g\circ f(3)$. If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\dfrac{x^2}{a}+\dfrac{y^2}{b}=1$, then $8e^2+l^2$ is equal to:
Step-by-Step Solution
Key Concept: $a=f(g(10))=f(10)=109$. $b=g(f(3))=g(18)=2$. Ellipse: $\frac{x^2}{109}+\frac{y^2}{2}=1$. $e^2=1-2/109=107/109$. $l=2b^2/a=4/\sqrt{109}$.
$a=109$, $b=2$. $8e^2+l^2=8(107/109)+16/109=872/109=8$.
Correct Answer: 1