Question 706690: the circle passes through the points A(-5,2) and B(-1,4) and tangent to the line x-5y=10.
1. radius of the circle
2. equation of a circle in a center-radius form
3. equation of circle in general form
Found 2 solutions by Alan3354, KMST:
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! the circle passes through the points A(-5,2) and B(-1,4) and tangent to the line x-5y=10.
1. radius of the circle
2. equation of a circle in a center-radius form
3. equation of circle in general form
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Find the eqn of the perpendicular bisector of the line thru A & B. The center is on that line.
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The intersection of the perp bisector and x-5y=10 is a point on the circle, point C.
Find the perp bisector of AC.
The intersection of the 2 perp bisectors is the center.
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The radius is the distance from the center to A, B or C.
Find the
Answer by KMST(5396)  (Show Source):
You can put this solution on YOUR website! I have a solution (two solutions actually), but there has to be a better way to get to the solution, and the results are so complicated that I suspect a typo in the problem.
The midpoint of AB is (-3,3).
The slope of the line through A and B is
The perpendicular bisector of AB is
 -->  --> 
The center of a circle passing through A and B has to be on that line, so its coordinates would be (h,k) with
The equation of the circle is
 where  is the radius.
I can even substitute and get as equation for our circle
Substituting the coordinates of B, I get an expression for 
 -->  -->  --> 
Now I can write the equation for our circle as
Simple. I just have to find  .
I know the circle is tangent to the line
 --> 
so that line and the circle have just 1 intersection point.
Substituting into the equation for the circle,
I can find that intersection point and more.
If that quadratic equation must have just one solution, the discriminant must be zero, so
The best I can do with that unwieldy equation is divide everything by 4 to get
Applying the quadratic formula
That can be simplified to
Substituting into , we get
 with  and
 with 
The approximate values give us points (-1.95,0.91) and (-72.49,141.98) for centers.
Exact values for are
 and 
and approximate values are  and  corresponding to the centers above, respectively.
 and 
The equation of the circles are
 for the small circle, and
 for the large circle.
Asking for the equation of circle in general form is cruel.
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