Moving circle and fixed circle x + y + 4y-32 = 0 are inscribed and pass through fixed point a (0,2). The trajectory equation of moving circle center P is obtained

Moving circle and fixed circle x + y + 4y-32 = 0 are inscribed and pass through fixed point a (0,2). The trajectory equation of moving circle center P is obtained

The circle x + y + 4y-32 = 0 is transformed into the standard equation, and the following results are obtained: x + (y + 2) = 36, center B (0, - 2), radius 6. Suppose the radius of the moving circle is r, center C (x, y), then r = AC, inscribed BC = 6-r, so AC + BC = 6, so it is an ellipse, AB is the intersection point, then C = 2,2a = 6, a = 3, B = 9-4 = 5, so x / 5 + Y / 9 = 1

It is known that the moving circle passes through the fixed point P (1,0) and is tangent to the fixed line ij: x = - 1. The m-equation of the locus of the center of the moving circle can be obtained from the point C above L

The trajectory of the moving circle obviously conforms to the definition of parabola: the ratio of the distance to the fixed point and the distance to the fixed line is equal to 1, so p / 2 = 1, 2p = 4
Therefore, the locus of the moving center is as follows:
y^2=4x