It is known that a and B are the right vertex and the upper vertex of the ellipse x ^ 2 / 36 + y ^ 2 / 9 = 1 respectively, and the moving point C moves on the ellipse. The trajectory equation of the center of gravity g of △ ABC is obtained, Is the parameter equation application inside!

Let the point coordinates of C on the ellipse (6,3)
A(6,0) B(0,3)
According to the gravity formula g (2 + 2cosa, 1 + Sina)
Let x = 2 + 2cosa, y = 1 + Sina
Resolve (X-2) ^ 2 + 4 (Y-1) ^ 2 = 4, you should understand

Taking the x-axis as the guide line, what is the trajectory of the parabolic focus of the vertex on the ellipse x ^ 2 / 4 + (Y-2) ^ 2?

Let the parabolic equation be (x-t) ^ 2 = 2p (y-p / 2);
Its vertex is (T, P / 2)
Then:
t^2/4 +(p/2-2)^2=1;
→t^2 +(p-4)^2 =4;
The focus of parabola is (T, P),
Then the trajectory equation is T ^ 2 + (P-4) ^ 2 = 4
Namely
x^2 +(y-4)^2 =4.

As shown in the figure, the right focus f (C, 0) of the ellipse Q: x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 passes through the point F. the moving straight line m rotates around the point F and intersects the ellipse at ab. two points P are the midpoint of the line ab
Finding the trajectory equation of point P

Let a (x1, Y1), B (X2, Y2), P (x, y) be substituted into the elliptic equation: X1 ^ 2 / A ^ 2 + Y1 ^ 2 / b ^ 2 = 1x2 ^ 2 / A ^ 2 + Y2 ^ 2 / b ^ 2 = 1, subtracting: (x1-x2) (x1 + x2) + (y1-y2) (Y1 + Y2) = 0 and: X1 + x2 = 2x, Y1 + y2 = 2Y, that is, (x1-x2) x + (y1-y2) y = 0, then the slope of AB k = (y1-y2) / (x1-x2) = - X / Y and K = (y-0)

It is known that the ellipse C: x ^ 2 / 8 + y ^ 2 = 1, the left focus f (- 2,0), if the line y = x + m intersects the ellipse C at two different points a and B, and the midpoint m of the line AB is in the circle
On x ^ 2 + y ^ 2 = 1, find the value of M

Centrifugation e = C / A, C = 2,2 / a = √ 2 / 2, a = 2 √ 2, B ^ 2 = a ^ 2-C ^ 2 = 4,
Let a (x1, Y1), B (X2, Y2),
The elliptic equation is: x ^ 2 / 8 + y ^ 2 / 4 = 1,
Y = x + M,
x^2/8+(x+m)^2/4=1,
3x^2+4mx+2m^2-8=0,
According to Veda's theorem,
x1+x2=-4m/3,
Let m (x0, Y0), x0 = (x1 + x2) / 2 = - 2m / 3,
y0=-2m/3+m=m/3,
If M is on the circle x ^ 2 + y ^ 2 = 1, then OP = 1,
x0^2+y0^2=1,
4m^2/9+m^2/9=1,
∴m=±3√5/5.

It is known that a and B are the right vertex and the upper vertex of the ellipse x ^ 2 / 36 + y ^ 2 / 9 = 1 respectively, and the moving point C moves on the ellipse. The trajectory equation of the center of gravity g of △ ABC is obtained, Is the parameter equation application inside!