Question

If the eccentricity of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1,(a > b)\) is \(\frac{2}{3}\) and its focal chord is \(3x+2y-6 = 0\), then the value of \(a^2+b^2\) is...

• A. \(11\)
• B. \(12\)
• C. \(13\)
• D. \(14\)

Hint:

A focal chord passes through a focus, which lies at (ae, 0).

Answer 1

The Correct Option is D

Step 1: Use the focus condition:
Put $y = 0$ in $3x + 2y = 6$: $x = 2$. So the focus $(ae, 0)$ has $ae = 2$.

Step 2: Solve:
$a = \frac{2}{e} = 3$, $b^2 = a^2 - a^2e^2 = 9 - 4 = 5$.
So $a^2 + b^2 = 14$.

Final Answer:
$a^2 + b^2 = 14$, option (D). \[ \boxed{14} \]