Question

Find the equation of the locus of the center of a moving circle tangent to the y-axis and to a circle with a radius 2 with center at (8, 2).

Answer 1

Let $P(x,y)$ be the center of a moving circle tangent to the y-axis and to a circle with a radius 2 with center at (8, 2).

The distance from point $P$ to the y-axis is $|x|$. The distance from the point $P$ to a center $(8, 2)$ is

$\sqrt{(x-8)^2+(y-2)^2}.$ The distance from the point $P$ to a circle is $\sqrt{(x-8)^2+(y-2)^2}-2.$

Then

$$\sqrt{(x-8)^2+(y-2)^2}-2=|x|$$

$$\sqrt{(x-8)^2+(y-2)^2}=|x|+2$$

$$x^2-16x+64+(y-2)^2=x^2+4|x|+4$$

$$(y-2)^2=16x+4|x|-60$$

Since $(y-2)^2\geq0, y\in \R,$ we take $|x|=x, x\geq0$

$$(y-2)^2=16x+4x-60$$

$$(y-2)^2=20x-60$$

The equation of the locus of the center of a moving circle tangent to the y-axis and to a circle with a radius 2 with center at (8, 2) is the equaion of the parabola

$$(y-2)^2=20x-60$$