Question

find the locus of the center of the circles of constant radius r which touches a given circle of radius rl (1) Externally (2) Internally, given rl >r....please answer it fast...

Answer 1

Let A(h,k) be the center of circle of radius r1 and B(x,y) be the center of circle of radius r as shown in the attachment.

When touches the circles externally,
Distance between A and B will be sum of the radius.
$\sqrt{ {(x - h)}^{2} + {(y - k)}^{2} } = r1 + r \\ squaring \: both \: sides \: and \: expanding \\ locus \: of \: center \: is \\ {x}^{2} + {y}^{2} - 2hx - 2ky + {h}^{2} + {k}^{2} - {(r1 + r}^{2} = 0$
When touches the circles internally,
Distance between A and B will be difference of the radius.
$\sqrt{ {(x - h)}^{2} + {(y - k)}^{2} } = r1 - r \\ squaring \: both \: sides \: and \: expanding \\ locus \: of \: center \: is \\ {x}^{2} + {y}^{2} - 2hx - 2ky + {h}^{2} + {k}^{2} - {(r1 - r}^{2} = 0$
Note: Both the locus of centers is circle