Question

A circle cuts a chord of length $4a$ on the $x$-axis and passes through a point on the $y$-axis, distant $2b$ from the origin. Then the locus of the centre of this circle, is :

• A. a straight line
• B. an ellipse
• C. a parabola
• D. a hyperbola

Answer 1

The correct option is C a parabola

Let the centre of the circle is $C(h,k)$ and radius $R$.

Now from $\Delta CAB$
$k^2+(2a{)}^2=R^2$
$\Rightarrow k^2+4a^2=R^2\cdots \left(1\right)$

$CD=R$
$\Rightarrow (h-0{)}^2+(k-2b{)}^2=R^2$
$\Rightarrow h^2+k^2+4b^2-4kb=R^2\cdots \left(2\right)$

Subtract equation $\left(1\right)$ from equation $\left(2\right)$
$h^2-4kb+4b^2-4a^2=0$
Replace $(h,k)$ with $(x,y)$
$\Rightarrow x^2-4yb+4b^2-4a^2=0$
$\Rightarrow (x^2+4b^2)=4(by+a^2)$
Hence, it is a parabola.