First slide

Definition of a circle

Question

A variable circle passes through the fixed point (2, 0) and touches the y-axis. Then, the locus of its centre, is

Moderate

A

a parabola
B

a circle
C

an ellipse
D

a hyperbola

Solution

Let the variable circle be
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ ... (i)
It passes through (2, 0) and touches y-axis.
$∴ 4 + 4 g + c = 0$ and $c = f^{2}$
$\Rightarrow 4 + 4 y + f^{2} = 0$
Hence, the locus of the centre $(- g, - f)$ of circle (i) is
$4 - 4 x + y^{2} = 0$
$\Rightarrow y^{2} = 4 (x - 1)$ , which is a parabola.

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