Given circle O: X & # 178; + Y & # 178; = 1 and point m (4,2) The equation for finding the straight line L through the tangent line L of the circle O in the direction of point m The equation for a circle m with chord length 4, which is cut by a straight line y = 2x-1 and centered on M Let p be any point in the middle circle m, passing through point P and leading tangent to circle O, and the tangent point is Q. try to explore whether there is a certain point R in the plane, so that PQ / PR is a fixed value? If there is, give an example. If not, explain the reason

Given circle O: X & # 178; + Y & # 178; = 1 and point m (4,2) The equation for finding the straight line L through the tangent line L of the circle O in the direction of point m The equation for a circle m with chord length 4, which is cut by a straight line y = 2x-1 and centered on M Let p be any point in the middle circle m, passing through point P and leading tangent to circle O, and the tangent point is Q. try to explore whether there is a certain point R in the plane, so that PQ / PR is a fixed value? If there is, give an example. If not, explain the reason

① Because the tangent passes through the point m, let the tangent equation be Y-2 = K (x-4), the distance from the center O to the tangent d = ｜ K · 0-0 + 2-4k / √ [K & # 178; + (- 1) &# 178;] = 1, k = (8 ± √ 3) / 15, and the equation of l be Y-2 = (8 ± √ 3) (x-4) / 15

Given that the center of circle C: (x-m) ² + (y-m + 1) ² = 5 is on the line 2x-y-1 = 0, then M=

m=0;
It is known from the equation that the center of the circle is (m, m-1), so 2m - (m-1) - 1 = 0 = > m = 0

It is known that the center of circle (x + 4) ^ 2 + y ^ 2 = 25 is m, and the center of circle (x-4) ^ 2 + y ^ 2 = 1 is m2. The moving circle is circumscribed with the two circles
1. Find the trajectory equation of moving circle center P
2. If there are two intersections A and B between the straight line passing through point m2 and the trajectory in (I), the value range of the solution is obtained
C = 4, how about a?

(x + 4) ^ 2 + y ^ 2 = (R + 5) ^ 2 = R ^ 2 + 10R + 25 (x-4) ^ 2 + y ^ 2 = (R + 1) ^ 2 = R ^ 2 + 2R + 116x = 8R + 242x = R + 3R = 2x-3 center track (x-4) ^ 2 + y ^ 2 = (x-1) ^ 8x ^ 2-8x + 16 + y ^ 2 = 8x ^ 2-16x + 87x ^ 2-8x-y ^ 2 = 8 ask whose value range is the second question

Given the circle C: X & # 178; + Y & # 178; + (2a + 1) x-ay-4 = 0. Find the trajectory equation of the center C

The formula is: [x + (2a + 1) / 2] ^ 2 + (Y-A / 2) ^ 2 = 4 + (2a + 1) ^ 2 / 4 + A ^ 2 / 4
The center of the circle is O (x, y), x = - (2a + 1) / 2 = - A-1 / 2, y = A / 2
Substituting a = - X-1 / 2, the equation of the center of the circle is a straight line
y=a/2=-x/2-1/4