Question

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
• D. A parabola

Answer 1

Answer: (D)

A parabola

Answer 2

Suppose the centre of circle be $( h , k )$. Since it touches the $y$-axis ,

$\therefore$radius of circle = h

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

$\Rightarrow$ $k ^ { 2 } = 4 h - 4$. Hence the locus of centre is $y ^ { 2 } = 4 x - 4$,

which is a parabola.