Circles
Circle
star_batch_jee_advanced_2025
Grade 11
Question:
A circle touches the hypotenuse of a right-angled triangle at its middle point and passes through the middle point of the shorter side. If $3$ units and $4$ units be the length of the sides and $'r'$ be the radius of the circle, then find the value of $'3r'$.
Step-by-Step Solution
Key Concept: The perpendicularity condition $CP \perp AB$ combined with equal distances $CP = CQ$ uniquely determines the centre.
Let the centre be $C(h, k)$ with $P = (3/2, 2)$ as midpoint of $AB$ and $Q = (3/2, 0)$ as midpoint of $OQ$. Using $CP \perp AB$ gives $6h - 8k = -7$. From $CP = CQ$, we get $(h - 3/2)^2 + (k - 2)^2 = (h - 3/2)^2 + k^2$. Solving yields $k = 1$ and $h = 1/6$. The radius is $r = CQ = \sqrt{(1/6 - 3/2)^2 + 1} = 5/3$, so $3r = 5$.
Correct Answer: 5