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 \ -\ 3) , 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:
}{2}\ =\ -8)
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:
^2\ +\ (y_1\ -\ y_2)^2})
)^2\ +\ (1\ -\ 5)^2})
Step 6: The equation of a circle with center at ) and radius  is ^2\ +\ (y\ -\ k)^2\ =\ r^2) , so read your values directly:
And the final form of the equation of your circle is:
^2\ +\ (y\ -\ 5)^2\ =\ 32)
John
My calculator said it, I believe it, that settles it
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