Question

已知圆O:x^2+y^2=9,过定点P(1,2)作互相垂直的两弦,AB,CD,则...

已知圆O:x^2+y^2=9,过定点P(1,2)作互相垂直的两弦,AB,CD,则线段AC的中点M的轨迹方程

Answer 1

设圆上两点为A(a,b),C(c,d),M(x,y):
a²
+
b²
=
9
(1)
c²
+
d²
=
9
(2)
x
=
(a
+
c)/2,a
+
c
=
2x
(3)
y
=
(b
+
d)/2,b
+
d
=
2y
(4)
AP²
+
CP²
=
AC²
(a
-
1)²
+
(b
-
2)²
+
(c
-
1)²
+
(d
-
2)²
=
(a
-
c)²
+
(b
-
d)²
简化得:a
+
c
+
2(b
+
d)
-
5
=
ac
+
bd
利用(3)(4):x
+
2y
-
5
=
ac
+
bd
(5)
直角三角形斜边上的中线等于斜边的一半:MP
=
AC/2
4MP²
=
AC²
4(x
-
1)²
+
4(y
-
2)²
=
(a
-
c)²
+
(b
-
d)²
=
a²
-
2ac
+
c²
+
b²
-
2bd
+
d²
=
(a²
+
b²)
+
(c²
+
d²)
-
2(ac
+
bd)
=
9
+
9
-
2(ac
+
bd)
利用(5)并整理得M的轨迹:
:(x
-
1/2)²
+
(y
-
1)²
=
13/4