Question

A circle passes through the points $A(1,0),B(5,0)$ and touches the y-axis at $C(0,h)$. If $\angle ACB$ is maximum then

• A. $h=3\sqrt{5}$
• B. $h=2\sqrt{5}$
• C. $h=\sqrt{5}$
• D. $h=2\sqrt{10}$

Answer 1

Answer: (C)

$h=\sqrt{5}$

Equation of any circles passing through $(1,0)$ and $(5,0)$ is
$y^2+(x-1)(x-5)+\lambda y=0$
i.e. $x^2+y^2+\lambda y-6x+5=0$
If $\angle$ACB is maximum, then this circle must touch the y-axis at $(0,h)$.

Putting $x=0$ in the equation of cirlce, we get $x^2+\lambda y+5=0$. It should have $y=h$ as it is a repeated root.
$\Rightarrow h^2=5$ and $\lambda =-2h$
$\Rightarrow |h|=\sqrt{5}$