Sum of circles equation 1. Let the circle C be tangent to the x-axis, inscribed with the square of the circle x + the square of the circle y + 4x-2y-76 = 0, and the radius be 4, then the equation of the circle C is obtained 2. Given that the equation of an internal bisector CD of triangle ABC is 2x + Y-1 = 0, and the two vertices are (1,2) B (- 1, - 1), find the coordinates of the third vertex C

Sum of circles equation 1. Let the circle C be tangent to the x-axis, inscribed with the square of the circle x + the square of the circle y + 4x-2y-76 = 0, and the radius be 4, then the equation of the circle C is obtained 2. Given that the equation of an internal bisector CD of triangle ABC is 2x + Y-1 = 0, and the two vertices are (1,2) B (- 1, - 1), find the coordinates of the third vertex C

The center of circle C is y = ± 4
Let the center of the circle be (a, 4)
（-2-a）²+（1-4）²=25
A = 2 or - 6
The equation of circle C;
(X-2) & sup2; + (y-4) & sup2; = 16 or (x + 6) & sup2; + (y-4) & sup2; = 16
Let the center of the circle be (a, - 4)
（-2-a）²+（1+4）²=25
a=2
The equation of circle C; (X-2) & sup2; + (y + 4) & sup2; = 16
The linear equation of AB: y = 3x / 2 + 1 / 2
The coordinates of the intersection point e with CD are (1 / 7,5 / 7)
Let C be (a, - 2A + 1)
AE/BE=AC/BC
[（1 - 1/7）²+(2-5/7)²]/[(1/7 +1)²+(5/7 +1)²=
[(a-1)²+(-2a+1-2)²]/[(a+1)²+(-2a+1+1)²]
35a²+86a-13=0
A = 1 / 7 or - 26
The coordinates of point C are (1 / 7,5 / 7) or (- 26,53)

On the equations of circle and line in the first year of senior high school
Given the eccentricity of ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > C), e = √ 6 / 3, the distance between the straight line passing through points a (0, - b) and B (a, 0) and the origin is √ 3 / 2. (1) the equation for solving ellipse (2) given the fixed point E (- 1,0), if the straight line y = KX + 2 (k is not equal to 0) intersects the ellipse at two points c and D, Q: is there a value of K to make the circle with diameter CD passing through point e? Please explain the reason
(1) It's solved. It's x ^ 2 / 3 + y ^ 2 = 1
(2) Can't write ~ help solve the second problem

Let the coordinates of CD be (x1, Y1), (X2, Y2) respectively
EC = (x1 + 1, Y1), ed = (x2 + 1, Y2), EC, ED are vectors
If e is on the circumference of a circle with diameter CD, then EC * ed = 0
(x1+1)(x2+1)+y1y2=0
x1x2+(x1+x2)+1+y1y2=0
x1x2+(x1+x2)+1+(kx1+2)(kx2+2)
(k²+1)x1x2+(2k+1)(x1+x2)+5=0
Substituting y = KX + 2 into elliptic equation
x²/3+(kx+2)²=1
(1/3+k²)x²+4kx+3=0
x1+x2=-4k/(1/3+k²),x1x2=3/(1/3+k²)
Simplify generation
3(k²+1)-4k(2k+1)+5(1/3+k²)=0
(14/3)-4k=0
k=7/6

The equation of circle and line in senior one
11. Find the equation of the circle C: X & # 178; + Y & # 178; - x + 2Y = 0 with respect to the symmetry of the line L: X-Y + 1 = 0

Let P (x, y) be any point on the circle,
The symmetric point Q (Y-1, x + 1) of point P with respect to line X-Y + 1 = 0 must be on circle C
There are (Y-1) & # 178; + (x + 1) & # 178; - (Y-1) + 2 (x + 1) = 0
We can get the equation of the circle
x²+y²+4x-3y+5=0

A question about the position relationship between circle and line in analytic geometry
It is known that circle C: (X-2) ^ 2 + (Y-3) ^ 2 = 4, line L: (M + 2) x + (2m + 1) y = 7m + 8. When the chord length of line L cut by circle C is the shortest, the value of M is obtained

Finishing line L
We get (x + 2y-7) m + (2x + Y-8) = 0
Solving equations
x+2y-7=0
2x+y-8=0
We get x = 3, y = 2
That is to say, the line L passes through the point P (3,2)
It is easy to know that point P is in circle C
To make the string shortest, l must be perpendicular to Op
The slope of OP is (2-3) / 3-2 = - 1
So the slope of L is 1
Then - (M + 2) / (2m + 1) = 1
M = - 1