document.write( "Question 706690: the circle passes through the points A(-5,2) and B(-1,4) and tangent to the line x-5y=10.
\n" ); document.write( "1. radius of the circle
\n" ); document.write( "2. equation of a circle in a center-radius form
\n" ); document.write( "3. equation of circle in general form \n" ); document.write( "

Algebra.Com's Answer #435477 by KMST(5398) $\"\"$ $\"About$

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.
\n" ); document.write( "The midpoint of AB is (-3,3).
\n" ); document.write( "The slope of the line through A and B is
\n" ); document.write( " $\"%284-2%29%2F%28-1-%28-5%29%29=2%2F4=1%2F2\"$
\n" ); document.write( "The perpendicular bisector of AB is
\n" ); document.write( " $\"y-3=-2%28x%2B3%29\"$ --> $\"y-3=-2x-6\"$ --> $\"y=-2x-3\"$
\n" ); document.write( "The center of a circle passing through A and B has to be on that line, so its coordinates would be (h,k) with
\n" ); document.write( " $\"k=-2h-3\"$
\n" ); document.write( "The equation of the circle is
\n" ); document.write( " $\"x-h%29%5E2%2B%28y-k%29%5E2=R%5E2\"$ where $\"R\"$ is the radius.
\n" ); document.write( "I can even substitute $\"k=-2h-3\"$ and get as equation for our circle
\n" ); document.write( " $\"x-h%29%5E2%2B%28y%2B2h%2B3%29%5E2=R%5E2\"$
\n" ); document.write( "Substituting the coordinates of B, I get an expression for $\"R%5E2\"$
\n" ); document.write( " $\"-1-h%29%5E2%2B%284%2B2h%2B3%29%5E2=R%5E2\"$ --> $\"h%5E2%2B2h%2B1%2B%287%2B2h%29%5E2=R%5E2\"$ --> $\"h%5E2%2B2h%2B1%2B4h%5E2%2B28h%2B49=R%5E2\"$ --> $\"5h%5E2%2B30h%2B50=R%5E2\"$
\n" ); document.write( "Now I can write the equation for our circle as
\n" ); document.write( " $\"x-h%29%5E2%2B%28y%2B2h%2B3%29%5E2=5h%5E2%2B30h%2B50\"$
\n" ); document.write( "Simple. I just have to find $\"h\"$ .
\n" ); document.write( "I know the circle is tangent to the line
\n" ); document.write( " $\"x-5y=10\"$ --> $\"x=5y%2B10\"$
\n" ); document.write( "so that line and the circle have just 1 intersection point.
\n" ); document.write( "Substituting $\"x=5y%2B10\"$ into the equation for the circle,
\n" ); document.write( " $\"x-h%29%5E2%2B%28y%2B2h%2B3%29%5E2=5h%5E2%2B30h%2B50\"$
\n" ); document.write( "I can find that intersection point and more.
\n" ); document.write( " $\"%285y%2B10-h%29%5E2%2B%28y%2B2h%2B3%29%5E2=5h%5E2%2B30h%2B50\"$
\n" ); document.write( " $\"25y%5E2%2B100y-10hy%2Bh%5E2%2B100-20h%2By%5E2%2B6y%2B4h%2B4h%5E2%2B9%2B12h=5h%5E2%2B30h%2B50\"$
\n" ); document.write( " $\"26y%5E2%2B%28106-6h%29y%2B5h%5E2%2B109-8h=5h%5E2%2B30h%2B50\"$
\n" ); document.write( " $\"26y%5E2%2B%28106-6h%29y%2B59-38h=0\"$
\n" ); document.write( "If that quadratic equation must have just one solution, the discriminant must be zero, so
\n" ); document.write( " $\"%28106-6h%29%5E2-4%2A26%2A%2859-38h%29=0\"$
\n" ); document.write( " $\"11236-1272h%2B36h%5E2-6136%2B3952h=0\"$
\n" ); document.write( " $\"36h%5E2%2B2680h%2B5100=0\"$
\n" ); document.write( "The best I can do with that unwieldy equation is divide everything by 4 to get
\n" ); document.write( " $\"9h%5E2%2B670h%2B1275=0\"$
\n" ); document.write( "Applying the quadratic formula
\n" ); document.write( " $\"h+=+%28-670+%2B-+sqrt%28+670%5E2-4%2A9%2A1275+%29%29%2F%282%2A9%29+\"$
\n" ); document.write( " $\"h+=+%28-670+%2B-+sqrt%28+448900-45900+%29%29%2F18+\"$
\n" ); document.write( " $\"h+=+%28-670+%2B-+sqrt%28403000%29%29%2F18+\"$
\n" ); document.write( "That can be simplified to
\n" ); document.write( " $\"h+=+%28-335+%2B-+5sqrt%284030%29%29%2F9+\"$
\n" ); document.write( "Substituting into $\"k=-2h-3\"$ , we get
\n" ); document.write( " $\"h+=+%28-335+%2B+5sqrt%284030%29%29%2F9+\"$ with $\"k=%28643-10sqrt%284030%29%29%2F9\"$ and
\n" ); document.write( " $\"h+=+%28-335+-+5sqrt%284030%29%29%2F9+\"$ with $\"k=%28643%2B10sqrt%284030%29%29%2F9\"$
\n" ); document.write( "The approximate values give us points (-1.95,0.91) and (-72.49,141.98) for centers.
\n" ); document.write( "Exact values for $\"R%5E2\"$ are
\n" ); document.write( " $\"R%5E2=%28978475-15400sqrt%284030%29%29%2F81\"$ and $\"R%5E2=%28978475%2B15400sqrt%284030%29%29%2F81\"$
\n" ); document.write( "and approximate values are $\"R=3.235\"$ and $\"R=155.4\"$ corresponding to the centers above, respectively.
\n" ); document.write( "

and

\n" ); document.write( "The equation of the circles are
\n" ); document.write( "

for the small circle, and
\n" ); document.write( "

for the large circle.
\n" ); document.write( "Asking for the equation of circle in general form is cruel. \n" ); document.write( "

\n" );