Question

If $\lambda x^2+\mu y^2+(\lambda \mu -4)xy-\lambda x-\mu y-20=0$ represents a circle, the radius of the circle is

• A. $\sqrt{\frac{21}{2}}$
• B. $\sqrt{\frac{42}{2}}$
• C. $2\sqrt{21}$
• D. $\sqrt{}22$

Answer 1

The correct option is A $\sqrt{\frac{21}{2}}$

The general equation of circle is $x^2+y^2=2gx+2fy+c=0$ where

$Center=(-g,-f)$ and

$Radius=\sqrt{g^2+f^2-c}$

So, for given equation equation to represent circle,

$\lambda =\mu$ and $\lambda \mu -4=0$ and radius should be real

Solving we get,

$\lambda =\mu =\pm 2$ (Taking $\lambda =-2$ gives non-real raidus so get rejected)

So, equation of circle,

$\to 2x^2+2y^2-2x-2y-20=0$

$\to x^2+y^2-x-y-10=0$

Now,

Radius = $\sqrt{(\frac{1}{2}{)}^2+(\frac{1}{2}{)}^2+10}=\sqrt{\frac{42}{4}}=\sqrt{\frac{21}{2}}$

Hence, (A) is the correct answer.