Question

The curve $x^2-y-\sqrt{5}x+1=0$ intersects $x-$axis at $A$ and $B.$ A circle is drawn passing through $A$ and $B.$ Then the length of the tangent drawn from the origin to the circle is

Answer 1

Given curve is parabola.
For point of intersection with $x-$axis, putting $y=0$
$x^2-\sqrt{5}x+1=0$
$\Rightarrow x=\frac{\sqrt{5}\pm 1}{2}$
$\therefore A\equiv (\frac{\sqrt{5}-1}{2},0)$ and $B\equiv (\frac{\sqrt{5}+1}{2},0)$

In the above figure,
$(OT{)}^2=OA\cdot OB$
$\Rightarrow (OT{)}^2=\left(\frac{\sqrt{5}-1}{2}\right)\left(\frac{\sqrt{5}+1}{2}\right)$
$\Rightarrow OT=1$