The locus of the centre of a circle, which touches externally the circle and also touches the y-axis, is given by the equation:
Step 1: Find the radius and centre of the given circle
In the question, a circle is given.
Rewrite the equation of the given circle as follows:
So, the given equation of circle can be rewritten as .
So, the centre of the given circle is and the radius is .
Step 2: Find the equation of the locus
Assume that, be the centre of a circle, which touches externally the given circle and also touches the y-axis.
Since the assumed circle touches the y-axis. so, its radius is h.
We know that the condition of circles to touch externally is: where is the distance between the centers of two circles and be the radii.
So, .
Taking square on both sides,
Now, replace h with x and k with y.
.
Therefore, the locus of the centre of a circle, which touches externally the circle and also touches the y-axis, is given by the equation: .
Hence, the correct answer is option D.