First slide

Definition of a circle

Question

The locus of the centre of a circle of radius 2 which rolls on the outside of the circle
$x^{2} + y^{2} + 3 x - 6 y - 9 = 0$ , is

Moderate

A

$x^{2} + y^{2} + 3 x - 6 y + 5 = 0$
B

$x^{2} + y^{2} + 3 x - 6 y - 31 = 0$
C

$x^{2} + y^{2} + 3 x - 6 y + \frac{29}{4} = 0$
D

none of these

Solution

Let $(h, k)$ be the coordinates of the centre of the circle which rolls on the outside of the circle
$x^{2} + y^{2} + 3 x - 6 y - 9 = 0$ . Then ,Distance between centres = Sum of the radii
$\begin{array}{l} \Rightarrow {(h + \frac{3}{2})}^{2} + (k - 3)^{2} = \frac{9}{2} + 2 \\ \Rightarrow h^{2} + k^{2} + 3 h - 6 k + \frac{9}{4} + 9 = \frac{9}{2} + 2 \\ \Rightarrow h^{2} + k^{2} + 3 h - 6 k - 31 = 0 \end{array}$
Hence, the locus of $f (h, k)$ is $x^{2} + y^{2} + 3 x - 6 y - 31 = 0$

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