Question

Centre of circles touching both axes and passing through $(-2,3)$ is

• A. $(5+2\sqrt{3},5+2\sqrt{3}),(5-2\sqrt{3},5-2\sqrt{3})$
• B. $(-5-2\sqrt{3},5+2\sqrt{3}),(-5+2\sqrt{3},5-2\sqrt{3})$
• C. $(5+2\sqrt{3},-5-2\sqrt{3}),(5-2\sqrt{3},-5+2\sqrt{3})$
• D. imaginary centre

Answer 1

The correct option is B $(-5-2\sqrt{3},5+2\sqrt{3}),(-5+2\sqrt{3},5-2\sqrt{3})$

It is clear that circle in $II$ quadrant
$\therefore$ Centre is $(-h,h),$
then radius$=\sqrt{{(-2+h)}^2+{(3-h)}^2}=\left|h\right|$
On simplifying, we get
$\Rightarrow 2h^2-10h+4+9=h^2$
$\Rightarrow h^2-10h+13=0$
$\therefore h=\frac{10\pm \sqrt{100-52}}{2}=5\pm 2\sqrt{3}$
Centres are $(-(5\pm 2\sqrt{3}),5\pm 2\sqrt{3})$
i.e., $(-5-2\sqrt{3},5+2\sqrt{3})$ and $(-5+2\sqrt{3},5-2\sqrt{3})$