Question

$(2+2\lambda +\mu )x^2+(1+2\lambda +2\mu )y^2+(3+5\lambda +3\mu )xy$

$-(16+14\lambda +11\mu )x-(10+13\lambda +17\mu )+(24+20\lambda +30\mu )=0$ is an equation of circle only when

• A. $\lambda =1\mu =1$
• B. $\lambda =-2\mu =\frac{-12}{11}$
• C. $\lambda =1\mu =0$
• D. $\lambda =\frac{-6}{5}\mu =1$

Answer 1

The correct option is D

$\lambda =\frac{-6}{5}\mu =1$

If given equation:

$(2+2\lambda +\mu )x^2+(1+2\lambda +2\mu )y^2+(3+5\lambda +3\mu )xy$

$-(16+14\lambda +11\mu )x-(10+13\lambda +17\mu )+(24+20\lambda +30\mu )=0$ is an equation of circle.

Then coefficient of $x^2=$coefficient of $y^2$

$2+2\lambda +\mu =1+2\lambda +2\mu$

$\mu =1$

And coefficient of $xy=0$

$3+5\lambda +3\mu =0$

Substituting $\mu =1$

$3+5\lambda +3=0$

$\lambda =\frac{-6}{5}$

So, option D is correct