Math CentralQuandaries & Queries


Question from Ladines a student:

Find the equation of the circle passing through points of intersections of circles x²+y²=4y and x²+y²=2x and the center is on line y=2

Hi Ladines,

I would first put the equations of the circles in the form

\[\left(x-h\right)^2 = \left(y-k\right)^2 = r^2\]

and then the center of the circle is $(h,k)$ and the radius of the circle is $r.$ To perform this you are going to have to complete the square.

For your first equation $x^2 + y^2 = 4y$ I would add $-4y$ to each side and then complete the square of $y^2 - 4y.$ From this you can read off the center and radius of the circle. Perform a similar task with the second equation.

If you look at the original equations you sent it is clear that $x = 0, y = 0$ satisfy both equations and hence $(0,0)$ lies on each graph.

At this point I would draw a reasonably accurate diagram of the two circles. What is the equation of the circle with center on the line $y = 2$ that passes through the points of intersection of the two circle in your diagram?


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