Question

The line $\lambda x+\mu y=1$ is a normal to a circle $2x^2+2y^2-5x+6y-1=0$ if

• A. $5\lambda -6\mu =2$
• B. $4+5\mu =6\lambda$
• C. $4+6\mu =5\lambda$
• D. none of these

Answer 1

The correct option is C $4+6\mu =5\lambda$

Since it the given line is a normal to the given circle, it must pass through the center of the given circle.

Therefore

$x^2+y^2-\frac{5}{2}x+3y-\frac{1}{2}=0$
$(x-\frac{5}{4}{)}^2+(y+\frac{3}{2}{)}^2-\frac{25}{16}-\frac{9}{4}-\frac{1}{2}=0$

Hence the center of the circle lies at $(\frac{5}{4},\frac{-3}{2})$

Substituting in the equation of the given line, we get

$\lambda \left(\frac{5}{4}\right)+\mu \left(\frac{-3}{2}\right)=1$
$5\lambda -6\mu =4$

$4+6\mu =5\lambda$