Question

If C is the centre of the ellipse $9x^2 + 16y^2 = 144$ and $S$ is one focus. The ratio of CS to major axis, is

• A. $\sqrt{7}:16$
• B. $\sqrt{7}:8$
• C. $\sqrt{7}:\sqrt{16}$
• D. $\sqrt{7}:4$

Answer 1

Answer: (B)

$C$ i.e., centre of the ellipse $9x^2 + 16y^2 = 144$ is $(0,0)$
Ellipse is $ \frac{x^{2}}{16}+\frac{y^{2}}{9} = 1 $
$\therefore a^{2} = 16, b^{2} = 9$
Since $b^{2} = a^{2}\left(1-e^{2}\right)$
$ \therefore 9 = 16\left(1-e^{2}\right)$
$\Rightarrow \frac{9}{16} = 1-e^{2} $
$\Rightarrow e^{2}= \frac{7}{16} $
$ \Rightarrow e= \frac{\sqrt{7}}{4 }$
$ \therefore $ Major axis $= 2a = 2\left(4\right) = 8$
$S$ is $ \left(ae, 0\right)= \left( 4 \frac{\sqrt{7}}{4 }, 0 \right) = \left( \sqrt{7}, 0\right)$
$\therefore CS = \sqrt{\left(7-0\right)^{2}+\left(0-0\right)^{2}} = \sqrt{7}$
$ \therefore CS : 2a = \sqrt{7}: 8 $