Question

Given $4{\lambda }^2-5{\mu }^2+6\lambda +1=0$. If $\lambda x+\mu y+1=0$ touches a definite circle, then centre and radius will be

• A. $(3,0)$ and $\sqrt{5}$
• B. $(-3,0)$ and $\sqrt{7}$
• C. $(0,3)$ and $\sqrt{5}$
• D. None of these

Answer 1

The correct option is A $(3,0)$ and $\sqrt{5}$

Let centre $=(\alpha ,\beta )$ and radius $=a$
Then $\frac{|\lambda \alpha +\mu \beta +1|}{\sqrt{{\mu }^2+{\lambda }^2}}=a$
$\Rightarrow ({\alpha }^2-a^2){\lambda }^2+({\beta }^2-a^2){\mu }^2+2\alpha \beta \lambda \mu +2\alpha \lambda +2\beta \mu +1=0$ ...(1)
$4{\lambda }^2-5{\mu }^2+6\lambda +1=0$ ...(2)
From (1) and (2)
$\frac{{\alpha }^2-a^2}{4}=\frac{{\beta }^2-a^2}{-5}=\frac{2\alpha \beta }{0}=\frac{2\alpha }{6}=\frac{2\beta }{0}=\frac{1}{1}$
$\Rightarrow {\alpha }^2-a^2=4,{\beta }^2-a^2=-5\Rightarrow \alpha =3,\beta =0\Rightarrow a=\sqrt{5}$
Centre $(3,0)$ and radius $=\sqrt{5}$