SOLUTION: I need your help The equation of a circle with radius is (x-h)^2+ (y-k)^2 if the center is at (h,k) Find the condition that a circle shall a. touch the x-axis b.have its center

Algebra ->  Circles -> SOLUTION: I need your help The equation of a circle with radius is (x-h)^2+ (y-k)^2 if the center is at (h,k) Find the condition that a circle shall a. touch the x-axis b.have its center      Log On


   

Question 771657: I need your help
The equation of a circle with radius is (x-h)^2+ (y-k)^2 if the center is at (h,k) Find the condition that a circle shall
a. touch the x-axis
b.have its center on the y-axis
c.touch both axes
d.have its center on the line 3x+4y=12
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The equation of a circle with radius r is %28x-h%29%5E2%2B+%28y-k%29%5E2=r%5E2 if the center is at %22%28h%2Ck%29%22

Each of the conditions sets (fixes) something about h, k and r, but one or two of those variables will still be free to take any value. The un-fixed variable(s) will still be in the equation, and the equation will represent an infinite number of circles.

a. If the circle touches the x-axis (is tangent to the x-axis), the center is at a distance r from the x-axis, at point %22%28h%2Cr%29%22
so the equation is highlight%28%28x-h%29%5E2%2B+%28y-r%29%5E2=r%5E2%29

b. If the circle has its center on the y-axis (which is the line x=0, the center is at %22%280%2Ck%29%22 and the equation is
highlight%28x%5E2%2B+%28y-k%29%5E2=r%5E2%29

c. If the circle touches both axes, the center is at a distance r from both axes, at point %22%28r%2Cr%29%22 and the equation is
highlight%28%28x-r%29%5E2%2B+%28y-r%29%5E2=r%5E2%29

d. If the circle has its center on the line 3x%2B4y=12 drawing%28200%2C200%2C-1%2C5%2C-1%2C5%2C%0D%0Agrid%280%29%2Cline%28-4%2C6%2C8%2C-3%29%29,
then
3h%2B4k=12 --> 4k=12-3h --> k=%2812-3h%29%2F4 --> k=3-%283%2F4%29h
and the equation gets ugly:
highlight%28%28x-h%29%5E2%2B+%28y-4%2B%283%2F4%29h%29%5E2=r%5E2%29