$ABCD$ is rectangle a circle passing through $C$ touches $AB$ and $AD$ at $M$ and $N$ respectively. If the perpendicular distance of $MN$ from $C$ is $5$ then the area of rectangle is ___.
Step-by-Step Solution
Key Concept: A circle tangent to two perpendicular sides of a rectangle (AB and AD at points M and N) has its center at (r,r) where r is radius. The chord MN lies on the line x+y=r, and the perpendicular distance from point C to this line equals 5, which constrains the rectangle dimensions through the relationship involving the circle's tangency conditions.
Let $r$ be the radius. The line $MN$ has equation $x + y = r$. The distance from center to line is $\frac{|a + b - r|}{\sqrt{2}} = 5$, giving $a + b = 5\sqrt{2} + r$. The circle $x^2 + y^2 - 2rx - 2ry + r^2 = 0$ contains points on the axes. Using $a^2 + b^2 = 2(5\sqrt{2} + r) - r^2 = 10\sqrt{2} + r^2$, the rectangle area is $ab = \frac{1}{2}[(a+b)^2 - (a^2 + b^2)] = \frac{1}{2}[(5\sqrt{2} + r)^2 - (10\sqrt{2} + r^2)] = 25$.
Correct Answer: 25