Circle
Optimization
IIT-JEE 2003
Grade 11
Question:
For the circle $x^2 + y^2 = r^2$, find the value of $r$ for which the area enclosed by the tangents from $P(6, 8)$ to the circle and the chord of contact is maximum. _____.
Step-by-Step Solution
Key Concept: Area $= \frac{r(100 - r^2)^{3/2}}{100}$ (derive via tangent length $\sqrt{100 - r^2}$ and chord of contact distance). Differentiate and set to zero: $100 - 4r^2 = 0 \Rightarrow r = 5$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 5