Question

The slope of the diameter of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, whose length is the $GM$ of the major and minor axes, is:

• A. $\sqrt{\frac{a}{b}}$
• B. $\sqrt{ab}$
• C. $\sqrt{\frac{b}{a}}$
• D. $\frac{a}{b}$

Answer 1

Answer: (C)

$\sqrt{\frac{b}{a}}$

The length of the major axis is $2a$ and the length of minor axis is $2b$

The diameter of ellipse is $\sqrt{2a\times 2b}=2\sqrt{ab}$

So the point on ellipse will be in the form of

$(\sqrt{ab}cos\theta ,\sqrt{ab}sin\theta )$

If we substitute it in the ellipse equation , we get $\frac{b}{a}{\cos }^2\theta +\frac{a}{b}{\sin }^2\theta =1$

$\Rightarrow cos\theta =\frac{\sqrt{a}}{\sqrt{a+b}}$

So we get $tan\theta =\frac{\sqrt{b}}{\sqrt{a}}$

Therefore the correct option is $C$