SOLUTION: Each of a set of circles is tangent to y=3 and to x^2+y^2=1. Find the equation satisfied by the coordinates of the centers of these circles.

Algebra ->  Formulas -> SOLUTION: Each of a set of circles is tangent to y=3 and to x^2+y^2=1. Find the equation satisfied by the coordinates of the centers of these circles.      Log On


   

Question 1130620: Each of a set of circles is tangent to y=3 and to x^2+y^2=1. Find the equation satisfied by the coordinates of the centers of these circles.
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


Let C(x,y) be the center of one of the circles that satisfy the given conditions.

The distance of C from the origin is sqrt%28x%5E2%2By%5E2%29

Every point on the given circle x^2+y^2=1 is 1 unit from the origin.

So the distance of C from the circle x^2+y^2=1 is sqrt%28x%5E2%2By%5E2%29-1

The distance of C from the line y=3 is 3-y

The required condition is that the distance of C from the circle x^2+y^2=1 be equal to the distance of C from the line y=3:

sqrt%28x%5E2%2By%5E2%29-1+=+3-y
sqrt%28x%5E2%2By%5E2%29+=+4-y
x%5E2%2By%5E2+=+%284-y%29%5E2%29
x%5E2%2By%5E2+=+y%5E2-8y%2B16
x%5E2+=+-8y%2B16
8y+=+-x%5E2%2B16
y+=+%28-1%2F8%29x%5E2%2B2

ANSWER: the equation of the locus of points that are centers of circles tangent to both x^2+y^2=1 and y=3 is

y+=+%28-1%2F8%29x%5E2%2B2