SOLUTION: Find the equation of the circle that passes through (1, 2) and (3,4); tangent to 4x - 3y

SOLUTION: Find the equation of the circle that passes through (1, 2) and (3,4); tangent to 4x - 3y - 2 = 0 at (-1, -2).

Question 1078608: Find the equation of the circle that passes through (1, 2) and (3,4); tangent to 4x - 3y - 2 = 0 at (-1, -2).
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Strange problem.
At first sight, there seems to be too much information.
Excess information may be useless, but consistent with the rest.
In this case, there is contradictory information.
As written, there is no solution.
Maybe there is a typo.
Maybe the circle is supposed to just be tangent to the line at (-1,-2),
and also pass through one other given point.

If the circle is tangent to any line at (-1,-2),
the circle passes through (-1,-2).
Three non-colinear points determine a circle.
In other words, given three points that are not on the same line,
there is one and only one circle that passes through all three.
So, there is only one circle that passes through
(1,2) , (3,4) and (-1,-2),
and that circle is not tangent to the line indicated.
That circle, with the tree points indicated,
and the line 4x-3y-2=0 is shown below.

I can make a circle go through (1,2) and (3,4) ,
and be tangent to the line 4x-3y-2=0,
but I cannot choose the point of tangency,
and it is not (-1,-2).

NOTE: There are a few things to know to find the equation of a circle given some points.
1) If you know two points on the circle,
the center of the circle is on the perpendicular bisector
of the line joining those two points.
2) If you are given a tangent and the point of tangency,
the center of the circle is on a line
perpendicular to the tangent
and passing through the point of tangency.
3) If a line through two points A and B on the circle,
and a tangent to the circle intersect at a point P,
the point of tangency, T, is located so that

4) with the center of the circle and a point on the circle,
you can find the radius (the distance between those points),
and have all the information needed to write the equation of the circle.
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