document.write( "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. \n" ); document.write( "
Algebra.Com's Answer #353239 by KMST(5328)\"\" \"About 
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\"x%5E2%2By%5E2=25\" is the equation of a circle with radius 5 centered at (0,0).
\n" ); document.write( "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.
\n" ); document.write( "However, we have two choices: the circles could be externally tangent or internally tangent.
\n" ); document.write( "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).
\n" ); document.write( "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.
\n" ); document.write( "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.
\n" ); document.write( " Do you see the right triangles with sides in the ratio 3-4-5 and hypotenuse length 3, 5, and 7?
\n" ); document.write( "The side ratios
\n" ); document.write( "vertical (colorful) leg/hypotenuse = 4/5 and
\n" ); document.write( "horizontal leg/hypotenuse = 3/5 allowed me to calculate the coordinates of the centers for the red and green circles.
\n" ); document.write( "The equation for the internally tangent red circle, centered at (9/5, 12/5) is \"%28x-9%2F5%29%5E2%2B%28y-12%2F5%29%5E2=2%5E2\".
\n" ); document.write( "The equation for the externally tangent green circle, centered at (21/5, 28/5) is \"%28x-21%2F5%29%5E2%2B%28y-28%2F5%29%5E2=2%5E2\". \n" ); document.write( "
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