Question

If $(\alpha ,\beta )$ is a point on the circle whose centre is on the x-axis and which touches the line $x+y=0$ at $(2,-2)$, then the greatest value of $\alpha$ is

• A. $4-\sqrt{2}$
• B. $6$
• C. $4+2\sqrt{2}$
• D. $4+\sqrt{2}$

Answer 1

The correct option is C $4+2\sqrt{2}$

Let $(a,0)$ is the centre C and P is $(2,-2)$ which is given
then $\angle COP={45}^o$
Since the equation of OP is $x+y=0$
thus also then $\angle OCP={45}^o$
$\therefore$ $OP=2\sqrt{2}=CP$. Hence, $OC=4$.
The point on the circle with the greatest x coordinates is B.
$\alpha =OB=OC+CB=4+2\sqrt{2}$