高中数学 点击放大
1)
a = 2, c = sqrt(3);
b = sqrt(a^2 -c^2) = 1;
标准方程：
x^2/4+y^2/1 = 1;
2)
设P (xp, yp), PA中点的坐标为 M(x, y);
x = (xp+1)/2, y = (yp+1/2)/2;
xp = 2x-1;
yp = 2y-1/2;
xp^2/4 + yp^2 = 1;
(2x-1)^2/4 + (2y-1/2)^2 =1;
(x-1/2)^2 + (y-1/4)^2/(1/4) = 1,
M的轨迹方程为：(x-1/2)^2 + (y-1/4)^2/(1/4) = 1;
3)
设 B坐标为 (xb, yb)...全部
1)
a = 2, c = sqrt(3);
b = sqrt(a^2 -c^2) = 1;
标准方程：
x^2/4+y^2/1 = 1;
2)
设P (xp, yp), PA中点的坐标为 M(x, y);
x = (xp+1)/2, y = (yp+1/2)/2;
xp = 2x-1;
yp = 2y-1/2;
xp^2/4 + yp^2 = 1;
(2x-1)^2/4 + (2y-1/2)^2 =1;
(x-1/2)^2 + (y-1/4)^2/(1/4) = 1,
M的轨迹方程为：(x-1/2)^2 + (y-1/4)^2/(1/4) = 1;
3)
设 B坐标为 (xb, yb), 则C点坐标为 (-xb, -yb);
BC的直线方程为： yb*x - xb*y = 0;
|BC|^2 = 4(xb^2 + yb^2), |BC| = 2*sqrt(xb^2+yb^2);
A到BC的距离是d = |xb/2-yb|/sqrt(xb^2+yb^2);
三角形ABC的面积 s = |BC|*d/2 = |xb/2-yb|，
s^2 = (xb/2-yb)^2 = xb^2/4 + yb^2 -xb*yb
=1 -2*(xb/2)*yb
<= 1 + (xb/2)^2 + yb^2
=2, 当且仅当 xb/2 +yb = 0 时取等号；
s<=sqrt(2);
故三角形ABC的面积最大值是 sqrt(2)。
。收起
