Question

Let S be the circle in xy-plane which touches the x-axis at point, A, the y-axis at point B and the unit circle $x^2+y^2=1$ at point C externally. If O denotes the origin, then the angle OCA equals.

• A. $\frac{5\pi }{8}$
• B. $\frac{\pi }{2}$
• C. $\frac{3\pi }{4}$
• D. $\frac{3\pi }{5}$

Answer 1

Answer: (A)

$\frac{5\pi }{8}$

Let the centre of circle $S$ be $P(a,a)$ and radius $=a$

Then, $OP=\sqrt{2}a$ (from Pythagoras theorem)

Also, $OP=\sqrt{2}+1+a$

$\sqrt{2}a=\sqrt{2}+1+a$

$\Rightarrow a=\frac{\sqrt{2}+1}{\sqrt{2}-1}=3+2\sqrt{2}$

Since $AO=AP,$

In $△OAP,$ $\angle AOP=\angle APO=\frac{\pi }{4}$

In $△PAC,$ $PA=PC=a$ and $\angle APC=\frac{\pi }{4}$

So, $\angle PAC=\angle PCA=\frac{3\pi }{8}$

Now, $\angle OCA=\pi -\angle PCA=\pi -\frac{3\pi }{8}=\frac{5\pi }{8}$