ABCD is a square of side length 2 units. `C_(1)` is the circle touching all the sides of the square ABCD and `C_(2)` is the circumcircle of square ABCD. L is a fixed line in the same plane and R is fixed point. If a circle is such that it touches the line L and the circle `C_(1)` externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is

Options

A. an ellipse
B. a hyperbola
C. a parabola
D. a pair of straight lines

Answer

C

Explanation

Let P be the centre of circle `C_(1)` of radius 1 unit and Q be the centre of the circle touching circle `C_(1)` externally and line L.

Draw a line `L_(1)` parallel to line L at a distance of 1 unit from L.
Produce QR to meet `L_(1)` at S.
Clearly, Q moves in such a way that it is equidistant from P and line `L_(1)`.
So, locus of Q is a parabola having focus at P and `L_(1)` as the directrix.