Question

Let $\pi (x,y)=0$ be the equation of a circle. If $\varphi (0,\lambda )=0$ has equal roots $\lambda =2,2$ and $\varphi (\lambda ,0)=0$ has roots $\lambda =\frac{4}{5},5$ then the centre of the circle is

• A. $(2,\frac{29}{10})$
• B. $(\frac{29}{10},2)$
• C. $(-2,\frac{29}{10})$
• D. None of these

Answer 1

The correct option is B $(\frac{29}{10},2)$

Given, $f(x,y)=0$ is the equation of a circle. And $f(0,\lambda )=0$ has equal roots $\lambda =2,2$, and $f(\lambda ,0)=0$ has roots $\lambda =\frac{4}{5},5$
Let $f(x,y)=x^2+y^2+2gx+2fy+c=0$ be the equation of the circle.
$f(0,\lambda )={\lambda }^2+2f\lambda +c=0$ whose roots are $2,2$
$\Rightarrow f=-2$ and $c=4$
$f(\lambda ,0)={\lambda }^2+2g\lambda +c=0$ whose roots are $\frac{4}{5},5$
$\Rightarrow g=-\frac{29}{10}$
$\therefore$ Centre of the circle $(-g,-f)=\left(\frac{29}{10},2\right)$
Hence, option B.