If the circle C passes through the coordinate origin and the center of the circle is on the straight line y = - 2x + 3, the equation of the circle with the minimum radius is obtained

If the circle C passes through the coordinate origin and the center of the circle is on the straight line y = - 2x + 3, the equation of the circle with the minimum radius is obtained

1. Distance formula method
Let p be a point on the straight line y = - 2x + 3, then the coordinates of P must satisfy: P (x, - 2x + 3)
The square of the distance from point P to the coordinate origin is: L ^ 2 = x ^ 2 + (- 2x + 3) ^ 2
L ^ 2 is the smallest, that is, l is the smallest (because l is always greater than 0)
L^2=x^2+4x^2-12x+9
=5x^2-12x+9
=5(x-1.2)^2+1.8
When x = 1.2, the minimum value of L ^ 2 is l (min) ^ 2 = 9 / 5 = 1.8, then p (1.2,0.6)
The circular equation is: (x-1.2) ^ 2 + (y-0.6) ^ 2 = 1.8
2. Straight line intersection method
When passing through the origin and straight line y = - 2x + 3, there must be y = 0.5x
(two vertical lines meet K1 * K2 = - 1)
The same results can be obtained when the intersection of two lines is (1.2,0.6)