<p>A circle touches the line L and the circle \(C_1\) externally such that both the circles are on the same side of the line. The locus of the centre of this circle is:</p>
Step-by-Step Solution
Key Concept: A parabola is the locus of points equidistant from a focus and directrix. Here, the distance from M to the circle centre minus the distance from M to the line equals a constant.
<p>Let the line L be the tangent line through A, and let the centre of the moving circle be at point M with radius \(r\).</p><p>Since the circle touches L, the distance from M to L equals \(r\).</p><p>Since the circle touches \(C_1\) externally, the distance from M to the centre of \(C_1\) equals \(r + 1\) (where 1 is the radius of \(C_1\)).</p><p>Therefore: distance from M to L = \(r\) and distance from M to centre of \(C_1\) = \(r + 1\).</p><p>This gives: distance from M to centre of \(C_1\) = distance from M to L + 1, which is the definition of a <strong>parabola</strong> with L as directrix and the centre of \(C_1\) as focus.</p>
Correct Answer: c