A variable circle passes through the fixed point (2,0) and touche… - Vidyadip

MCQ

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

A
A circle
B
An Ellipse
C
A hyperbola
✓
A parabola

✓

Answer

Correct option: D.

A parabola

d
(d) Suppose the centre of circle be $(h,k)$. Since it touches the $y{\rm{ - axis}}$

$\therefore $ radius of circle = $h$

Now ${(h - 2)^2} + {k^2} = {h^2}$

$ \Rightarrow $${h^2} + 4 - 4h + {k^2} = {h^2}$

$ \Rightarrow $ ${k^2} = 4h - 4$.

Hence the locus of centre is ${y^2} = 4x - 4$, which is a parabola.

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