SOLUTION: I'm stuck on a question, please help!! :) " Determine the equation of a circle with a diameter whose endpoints are P(5,-6) and Q(1,2)" I used the midpoint formula and got (3

Algebra ->  Circles -> SOLUTION: I'm stuck on a question, please help!! :) " Determine the equation of a circle with a diameter whose endpoints are P(5,-6) and Q(1,2)" I used the midpoint formula and got (3      Log On


   

Question 397031: I'm stuck on a question, please help!! :)
" Determine the equation of a circle with a diameter whose endpoints are P(5,-6) and Q(1,2)"
I used the midpoint formula and got (3,-2) for the midpoint, and then sketched out the graph and ended up with 8 for the hypotenuse and 4 for the other side. So then I went
"PQ=4^2+8^2=16 =16+64=80 "
and then went
" PQ = sqrt ( 80 ) "
And now I'm stuck! How do I finish the question? And am I even doing this right??
Found 2 solutions by jim_thompson5910, robertb:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You find the midpoint, which is the center of the circle. Since you found the midpoint to be (3,-2), this means that the center of the circle is (3,-2). This is only possible because the line segment PQ is the diameter.


Recall that the equation of the circle with radius 'r' and center (h,k) is %28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2


So because the center of the circle is (3,-2), this means that h=3 and k=-2. Plug these values into the equation above to get


%28x-3%29%5E2%2B%28y-%28-2%29%29%5E2=r%5E2


and simplify to get


%28x-3%29%5E2%2B%28y%2B2%29%5E2=r%5E2


So all we need to find out is the value of r%5E2. So we need to find the value of the radius 'r'. It turns out that the distance from the center (3,-2) to either point will be equal to the radius (draw a picture if you're not sure)


So to find 'r', let's find the distance from P(5,-6) to the center (3,-2)


Note: is the first point . So this means that x%5B1%5D=5 and y%5B1%5D=-6.
Also, is the second point . So this means that x%5B2%5D=3 and y%5B2%5D=-2.


d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 Start with the distance formula.


d=sqrt%28%285-3%29%5E2%2B%28-6--2%29%5E2%29 Plug in x%5B1%5D=5, x%5B2%5D=3, y%5B1%5D=-6, and y%5B2%5D=-2.


d=sqrt%28%282%29%5E2%2B%28-6--2%29%5E2%29 Subtract 3 from 5 to get 2.


d=sqrt%28%282%29%5E2%2B%28-4%29%5E2%29 Subtract -2 from -6 to get -4.


d=sqrt%284%2B%28-4%29%5E2%29 Square 2 to get 4.


d=sqrt%284%2B16%29 Square -4 to get 16.


d=sqrt%2820%29 Add 4 to 16 to get 20.


So this distance is sqrt%2820%29 units. So r=sqrt%2820%29


This then means that r%5E2=%28sqrt%2820%29%29%5E2=20


So r%5E2=20


Plug this last value into the equation from the beginning to get


%28x-3%29%5E2%2B%28y%2B2%29%5E2=20



So the equation of the circle with a diameter whose endpoints are P(5,-6) and Q(1,2) is %28x-3%29%5E2%2B%28y%2B2%29%5E2=20


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
You are right that PQ = sqrt%2880%29+=+4sqrt%285%29. Remember that this is the LENGTH OF THE DIAMETER, and so to get the radius you only have to divide this by 2:
Radius = +%284sqrt%285%29%29%2F2+=+2sqrt%285%29
The equation of the circle is then %28x-3%29%5E2+%2B+%28y+%2B+2%29%5E2+=++%282sqrt%285%29%29%5E2+=+20.