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 the circle is

• A. A hyperbola
• B. A parabola
• C. A straight line
• D. An ellipse

Answer 1

The correct option is B A parabola

Let equation of circle is

$x^2+y^2+2fx+2fy+e=0,itpassesthrough(0,2b)$

$⟹0+4b^{2}2g\times 0+4f+c=0$

$⟹4b^2+4f+c=0......\left(i\right)$

$2\sqrt{g^2-c=4a}$ ......(ii)

$g^2-c=4a^2⟹C=(g^2-4a^2)$

Putting in equation (1)

$⟹4b^2+4f+g^2-4a^2=0$

$⟹x^2+4y+4(b^2-a^2)=0$, it represent a parabola