Finding the equation of the circle given two points and a tangent line

AI Thread Summary
To find the equation of a circle that passes through the point (3,-2) and is tangent to the line y=3x+5 at (-1,2), the center of the circle must be determined. The radius can be calculated using the distance formula, and the standard form of the circle's equation is (x-h)² + (y-k)² = r². If calculus is applicable, three equations can be derived from the tangent point and the circle's intersection with the given point. Alternatively, if calculus is not an option, the center can be found by determining the perpendicular line to the tangent at (-1,2) and locating the point equidistant from the two given points. This approach will lead to the correct equation of the circle.
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Homework Statement


Find the equation of the circle that passes through the point (3,-2) and tangent to the line y=3x+5 at (-1,2). Answer in standard form.


Homework Equations


d= |mx0+b-y0|
____________
sqrt(1+m2

is needed to find the radius of the circle
(x-h)2+(y-k)2=r2 is needed to find the equation of the circle.

The Attempt at a Solution



I haven't attempted a solution per se, but I do know that I need to find the center of the circle. However I don't see a way to find the center of a circle.
 
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You need to figure out h, k, and r, right? Three unknowns. Here are a couple of suggestions, depending on whether you know any calculus.

If you know calculus, you can get three equations in the three unknowns by using the fact that your two given points must work in the equation, and the slope of the circle must equal the slope of the tangent line where they touch.

If you can't use calculus, write the equation of the line perpendicular to the tangent line at (-1,2) and find the point on that line that is equidistant from your two given points. That will be your center.
 

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