Question

In the figure shown, if the bigger circle has radius 1 unit, then the radius of the smaller circle must be

• A. $\sqrt{2}+1$
• B. $\frac{1}{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{1}{\sqrt{2}+1}$

Answer 1

The correct option is D $\frac{1}{\sqrt{2}+1}$

Let the center of the bigger circle be origin.

Equation of bigger circle is $x^2+y^2=1^2$ . Let $r$ be the radius of the smaller circle.

Equation of smaller circle that touches both the axes is

$(x-r{)}^2+(y-r{)}^2=r^2$

Since the circles touch internally

$C_1{C}_2=|r_1–r_2|$

$\sqrt{r^2+r^2}=1-r$

$\sqrt{2}r=1-r$

$⟹r=\frac{1}{\sqrt{2}+1}$