It is known that the parabola y = x ^ 2-2 and the ellipse x ^ 2 / 4 + y ^ 2 = 1 have four intersections If the four points are in a circle, the equation of the circle is

It is known that the parabola y = x ^ 2-2 and the ellipse x ^ 2 / 4 + y ^ 2 = 1 have four intersections If the four points are in a circle, the equation of the circle is

A simple calculation method is provided
y=x^2-2
x^2=y+2
x^2/4+y^2=1
(y+2)/4+y^2=1
4Y ^ 2 + y = 2 (in duplicate)
Because the four points are in the same circle, according to the image, the center of the circle is on the y-axis. Let the center coordinate be (0, P) and the radius be r
Then the circular equation is x ^ 2 + (y-p) ^ 2 = R ^ 2
Substitute x ^ 2 = y + 2 to get y + 2 + (y-p) ^ 2 = R ^ 2
Y ^ 2 + (1-2p) y = R ^ 2-2-p ^ 2
Because these four points all satisfy the equation of one form and two forms, two different Y values can be generated in these four points, and each quadratic equation can generate two Y values, so the two equations are actually the same, that is, after multiplying the left and right of two forms by four, they are actually in one form, so
4(1-2p)=1,p=3/8
4r^2-8-4p^2=2
r=13/8
The circular equation is x ^ 2 + (Y-3 / 8) ^ 2 = 169 / 64
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