SOLUTION: A circle passes through the Point (8,7) and touches the y-axis at the Point (0,3). Find the equation of the circle. if the circle cuts the x-axis at D and E, find the equation of a

Algebra ->  Circles -> SOLUTION: A circle passes through the Point (8,7) and touches the y-axis at the Point (0,3). Find the equation of the circle. if the circle cuts the x-axis at D and E, find the equation of a      Log On


   

Question 990255: A circle passes through the Point (8,7) and touches the y-axis at the Point (0,3). Find the equation of the circle. if the circle cuts the x-axis at D and E, find the equation of another circle which is DE as diameter.

Answer by josgarithmetic(39625) About Me  (Show Source):
You can put this solution on YOUR website!
Unknown center of the first circle, (h,k).
Using Distance Formula
%28h-8%29%5E2%2B%28k-7%29%5E2=%28h-0%29%5E2%2B%28k-3%29%5E2
%28h-8%29%5E2%2B%28k-7%29%5E2=h%5E2%2B%28k-3%29%5E2
h%5E2-16h%2B64%2Bk%5E2-14k%2B49=h%5E2%2Bk%5E2-6k%2B9
-16h%2B64-14k%2B49=-6k%2B9
-16h-14k%2B6k%2B64%2B49-9=0
-16h-8k%2B64%2B40=0
16h%2B8k=104
2h%2Bk=13

This circle touching, just touching, the y-axis at (0,3) means that the center is some point on the line y=3. This means, you can solve for h, because you know k=-3. This is based on knowing how the standard circle equation works.

2h=13-k
2h=13-%28-3%29
2h=16
h=8

Center of this first circle is therefore, (8,-3). The point (0,3) on the circle may be the convenient point to again use the Distance Formula, to find the radius of this circle.

r%5E2=%288-0%29%5E2%2B%28-3-0%29%5E2
r%5E2=8%5E2
r=8

This first circle equation is then, highlight%28%28x-8%29%5E2%2B%28y%2B3%29=8%29.
I have not finished to do the final question, but maybe you can.