SOLUTION: Can someone help me with this question please? A circle C passes through the point (-12, 9) and is given by the equation {{{ (x-a)^2 + (y-b)^2 = r^2 }}} If the equation of t

Question 597160: Can someone help me with this question please?
A circle C passes through the point (-12, 9) and is given by the equation

If the equation of the tangent to the given circle at the point (-4, 1) is given by:

Find the values of a,b and r.
Thank you.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

This looks a good deal uglier than it actually is. The fortunate placement of the the point (-12,9) makes things much easier, as you will see.

Step 1: Take the given tangent line equation and put it into slope-intercept form:

noting that the slope is 1.

Step 2: Use the fact that a radius to a point of tangency is perpendicular to the tangent line at that point, perpendicular lines have negative reciprocal slopes, and the point-slope form to write an equation of the line containing the radius from the center of the circle to the point of tangency.

Step 3: Here is the fortunate circumstance. Since

, the point (-12, 9) is the other end of the diameter contained in the line

. Therefore the center of the circle must be the midpoint of the segment with endpoints (-12, 9) and (-4, 1)!

Step 4: Calculate the midpoint coordinates using the midpoint formulas:

Hence the midpoint of the diameter which must be the center of the circle is (-8, 5).

Step 5: The distance from either endpoint to the center is the radius, so using the distance formula: