Question

If the circles touching both axes and passing through $(2,3)$ then the length of the common chord is

• A. $\sqrt{2}$
• B. $2\sqrt{2}$
• C. $3\sqrt{2}$
• D. $4\sqrt{2}$

Answer 1

The correct option is A $\sqrt{2}$

From the figure, $tan\theta =\frac{\frac{3}{2}-1}{1+\frac{3}{2}}=\frac{1}{5}$

In $\Delta ADO,sin\theta =\frac{AD}{OA}$

We have $tan\theta =\frac{1}{5}$

$\Rightarrow \frac{sin\theta }{cos\theta }=\frac{1}{5}$

$5sin\theta =cos\theta$

using identity $si{n}^{2}x+co{s}^{2}x=1$ we get

$si{n}^2\theta +25si{n}^2\theta =1$

$\Rightarrow 26si{n}^2\theta =1$

$\therefore sin\theta =\frac{1}{\sqrt{26}}$

$\Rightarrow \frac{1}{\sqrt{26}}=\frac{AD}{OA}$

$\Rightarrow \frac{1}{\sqrt{26}}=\frac{AD}{\sqrt{13}}$

$\Rightarrow AD=\frac{\sqrt{13}}{\sqrt{26}}$

$\Rightarrow AD=\frac{1}{\sqrt{2}}$

Thus the length of the common chord is $2AD=2\times \frac{1}{\sqrt{2}}=\sqrt{2}$