Question 1146375
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The required conditions for each point in the locus are<br>
(1) The radius of the circle is the y value; and
(2) The radius of the circle plus 4 is the distance between the point and the center of the circle.<br>
Those two requirements give us, for any point (x,y) on the locus,<br>
{{{y+4 = sqrt((x-6)^2+(y-8)^2))}}}
{{{y^2+8y+16 = x^2-12x+36+y^2-16y+64}}}
{{{0 = x^2-12x-24y+84}}}
{{{24y = (x^2-12x+36)+48}}}
{{{y = (1/24)(x-6)^2+2}}}<br>
Here is another path to the same solution.<br>
With the circle with center (6,8) and radius 4, the point on the circle closest to the x-axis is (6,4); then clearly one point on the locus is (6,2) -- halfway between (6,4) and the x-axis.<br>
It should also by clear by symmetry that the locus will be symmetrical about the line x=6; any point on the locus "a" units to the right of the line x=6 will be mirrored by a point on the locus "a" units to the left of the line x=6.<br>
So the locus is symmetrical about the line x=6; it is therefore a parabola with vertex (6,2) and an equation of the form<br>
{{{y-2 = a(x-6)^2}}}<br>
To determine the value of the constant a, note that a point on the locus to the right of x=6 will have a y value of 8 and an x value of 6 (center of circle) plus 4 (radius of circle) plus 8 (equal to the y value).<br>
That gives us the point (18,8).  Plugging those values in the equation of the parabola gives us the value of a:<br>
{{{8-2 = a(18-6)^2}}}
{{{6 = 144a}}}
{{{a = 1/24}}}<br>
And again we find the equation of the locus i<br>
{{{y-2 = (1/24)(x-6)^2}}}<br>
or<br>
{{{y = (1/24)(x-6)^2+2}}}