Question

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

• A. $(2,\frac{29}{10})$
• B. $(\frac{29}{10},2)$
• C. $(-2,\frac{29}{2})$
• D. non of these

Answer 1

The correct option is B

$(\frac{29}{10},2)$

Determine the center of the circle

$\lambda =\frac{5}{4},5$ are the roots of $f(\lambda ,0)=0$
Consider $f(x,y)=x^2+y^2+2gx+2fy+c=0$the equation of the circle.
$\Rightarrow$$f(0,\lambda )={\lambda }^2+2f\lambda +c=0$whose roots are$2,2$ [Given]

On solving we get,$f=-2$ and $c=4$
$\Rightarrow$$f(\lambda ,0)={\lambda }^2+2g\lambda +c=0$ whose roots are $\frac{5}{4},5$ [Given]

On solving we get, $g=-1029$
We know that the center of a circle is given by : $(-g,-f)=(\frac{10}{29},2)$
Therefore the correct option is B.