If a circle touches the hypotenuse of a right-angled triangle at its middle point and passes through the middle point of shorter side. If $a$ and $b$ $(a < b)$ be the length of the sides and the radius of the circle is $\frac{b}{ka}\sqrt{a^2 + b^2}$, then the value of $k$ is __________.
Step-by-Step Solution
Key Concept: Set up a coordinate system with the right angle at origin, legs along axes of lengths a and b. Use the constraint that the circle touches the hypotenuse at its midpoint (which lies on the line ax + by = ab) and passes through (a/2, 0), then equate the radius expressions to find k.
The circle passing through point $\left(\frac{a}{2}, 0\right)$ with center $\left(\frac{a}{2}, \frac{b}{2}\right)$ gives the parameter $\lambda = \frac{b}{2a}$. The resulting circle equation simplifies to $x^2 + y^2 + x\left(-a + \frac{b^2}{2a}\right) + y\left(\frac{b}{2} - b\right) + \frac{a^2 - b^2}{4} = 0$ with radius $\frac{b}{2a}\sqrt{a^2 + b^2}$.
Correct Answer: 4