Circles
Circle
Allen Star Batch
Grade 11
Question:
If $A(0,a)$ and $B(0,\beta)$, $a, \beta > 0$ are two vertices of a variable triangle $ABC$, where the vertex $C(x, 0)$ is variable. The value of $x$ for which $\angle ACB$ is maximum is:
$\frac{a+\beta}{2}$
$\sqrt{a\beta}$
$\frac{2a\beta}{a+\beta}$
$\frac{a\beta}{a+\beta}$
Step-by-Step Solution
Key Concept: Maximizing the angle subtended by a chord requires the circle to be tangent to a given line.
For $\triangle ACB$ to be maximum, the circle passing through $A$ and $B$ must touch the $x$-axis at $C$. Using the power of point $O$: $OC^2 = (OA)(OB)$, so $x^2 = a\beta$, giving $x = \sqrt{a\beta}$.
Correct Answer: 2