CircleHardBloom L3
Question
$ABCD$ is a square of unit area. A circle is tangent to two sides of $ABCD$ and passes through exactly one of its vertices. The radius of the circle is:
Options
A.$2 - \sqrt{2}$
B.$\sqrt{2} - 1$
C.$\frac{1}{2}$
D.$\frac{1}{\sqrt{2}}$
Solution
[Key Concept: Unit square, circle tangent to two sides and through a vertex: get the radius from the distance relation.]
Equation of circle is
$$x^{2} + y^{2} - 2rx - 2ry + r^{2} = 0$$
Put $(1,1)$
$$\begin{matrix} & \ \Rightarrow \ r^{2} - 4r + 2 = 0,r = 2 \pm \sqrt{2} \\ & \ \Rightarrow \ r = 2 - \sqrt{2}\ (\because r < 1) \end{matrix}$$
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