Question

The locus of the centres of the circles, which touch the circle, $x^2+y^2=1$ externally, also touch the $y-$axis and lie in the first quadrant, is:

• A. $y=\sqrt{1+2x},x\ge 0$
• B. $y=\sqrt{1+4x},x\ge 0$
• C. $x=\sqrt{1+4y},y\ge 0$
• D. $x=\sqrt{1+2y},y\ge 0$

Answer 1

The correct option is A $y=\sqrt{1+2x},x\ge 0$

Let the centre of the circle be $(h,k)$


Now, distance between the centers of the circles is
$\sqrt{h^2+k^2}=1+h$
squaring both sides
$h^2+k^2=1+h^2+2h$
$k^2=1+2h$
Replace $k$ with $y$ and $h$ with $x$
$y^2=1+2x$
$\Rightarrow y=\sqrt{1+2x},x\ge 0$