Circles
Circle
Allen Star Batch
Grade 11
Question:
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: Determining the parameter $\lambda$ from the passing-through condition allows full circle specification and radius calculation.
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