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Prove that the locus of the center of th...

Prove that the locus of the center of the circle which touches the given circle externally and the given line is a parabola.

A

circle

B

line

C

parabola

D

ellipse

Text Solution

Verified by Experts

The correct Answer is:
C

Without the loss of generality, the circle can be taken as `x^(2) + y^(2) = a^(2)` and the given line as `x =b =0`.
Let (h,k) be the centre of the required circle. Then length of tangent from (h,k) to the circle and distance of (h,k) from the line should be equal.
Hence, `sqrt(h^(2) + k^(2) -a^(2)) = |h-b|` or `k^(2) -a^(2) =- 2bh +b^(2)`
Thus, the locus is `y^(2) - a^(2) =- 2bx +b^(2)`.
i.e, `y^(2) =- 2bx +b^(2) +a^(2)` which is a parabola.
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