Question

Let $\varphi (x,y)=0$ 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 roots of the circle is

• A. (2, 29/100)
• B. (29/10, 2)
  
• C. (-2, 29/10)
  
• D. none of these

Answer 1

The correct option is B

(29/10, 2)

Let $\varphi (x,y)\equiv x^2+y^2+2gx+2fy+c=0\therefore \varphi (0,\lambda )=0+{\lambda }^2+0+2f\lambda +c=0$
Have equal roots,
$\text{Then}2+2=-\frac{2f}{1}\text{and}2.2=\frac{c}{1}\therefore f=-2\text{and}c=4\text{and}\varphi (\lambda ,0)\equiv {\lambda }^2+0+2g\lambda +0+c+0\therefore {\lambda }^2+2g\lambda +c=0\text{Here}c=4\therefore {\lambda }^2+2g\lambda +4=0\text{have roots}4/5,5\therefore \frac{4}{5}+5=-2g\Rightarrow g=-\frac{29}{10}$

$\therefore Centre\equiv (-g,-f)=(\frac{29}{10},2)$