Question 538672: Find the equation of the circle of radius 2 that contains point (3,4) and is tangent to x^2+y^2=25.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  is the equation of a circle with radius 5 centered at (0,0).
The point (3,4) belongs to that circle and is the only point in common with the tangent circle(s). The circles are tangent at that point.
However, we have two choices: the circles could be externally tangent or internally tangent.
In either case, at (3,4), the radius of the given circle and the radius of the tangent circle are on the same line, a line that passes through (0,0) and (3,4). The center of the tangent circle(s) is (are) on that line at a distance of 2 from (3,4).
We could find the coordinates of the center(s) in different ways. I can think of solving equations involving distance, invoking sine and cosine of the angle of that radius with the positive x-axis, or considering similar right triangles.
I used the last option to find centers at (9/5, 12/5) and at (21/5, 28/5). The centers are at the top of the red and green lines in the figure below.
 Do you see the right triangles with sides in the ratio 3-4-5 and hypotenuse length 3, 5, and 7?
The side ratios
vertical (colorful) leg/hypotenuse = 4/5 and
horizontal leg/hypotenuse = 3/5 allowed me to calculate the coordinates of the centers for the red and green circles.
The equation for the internally tangent red circle, centered at (9/5, 12/5) is  .
The equation for the externally tangent green circle, centered at (21/5, 28/5) is  .
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