Question

Let $(a,b)$ be a point on a circle which passes through $(-3,1)$ and touches the line $x+y=2$ at the point $(1,1).$ If maximum possible value of $a$ is $\alpha ,$ then a quadratic equation with rational coefficients whose one root is $\alpha ,$ is

• A. $x^2+2x-9=0$
• B. $x^2+2x-7=0$
• C. $x^2+2x-6=0$
• D. $x^2+2x-8=0$

Answer 1

The correct option is B $x^2+2x-7=0$

Equation of the circle touching the line $x+y-2=0$ at the point $(1,1)$ is given by
$S:(x-1{)}^2+(y-1{)}^2+\lambda (x+y-2)=0$
As $S$ passes through the point $(-3,1),$
$(-4{)}^2+0^2+\lambda (-3+1-2)=0\Rightarrow \lambda =4$

$\therefore S:x^2+y^2+2x+2y-6=0$
Centre $\equiv (-1,-1)$ and $r=\sqrt{(-1{)}^2+(-1{)}^2+6}=2\sqrt{2}$
If $(a,b)$ is the point on the circle, then the maximum value of $a$ is $-1+2\sqrt{2}.$
So, the other root of the quadratic equation will be $-1-2\sqrt{2}.$
Hence, required quadratic equation is
$x^2-(-2)x+(-1{)}^2-(2\sqrt{2}{)}^2=0\Rightarrow x^2+2x-7=0$