math T sem 1第五课的问题

Choon96

以下是pelangi里面的练习，我试了很多次都做不到，请问有人会吗？

1. find the locus of a point which is equidistant from A(4,4) and B(-4,-1). show that it represents the equation of the perpendicular bisector of the line AB.(这题我只会做前面的罢了，后面的不懂怎样做.)

2. if the normal at P(ap^2, 2ap) to the parabola y^2=4ax meets the curve again at Q(aq^2, 2aq), show that p^2 +pq +2=0. show that the equation of the locus of the point of intersection of the tangent at P and Q to the parabola is y^2(x+2a)+4a^3=0.

3. the line y=mx+c intersects the parabola y^2=4ax at the points P and Q. show that the coordinates of the mid-point of PQ is ( (2a-mc)/m^2 , 2a/m). if this mid-point is M, find the locus of M when m varies and c=1.

4. P and Q are two variable points lying on the hyperbola x=3t, y=3/t. the tangents at P and Q meet at T. if PQ passes through the point(6,2), find the equation of the locus of T as PQ varies.

5. the line 2y=x+7 intersects the curve x=2t, y=2/t at A and B. find the respective values of t corresponding to A and B. hence, find the coordinates of the point of intersection of the tangents at A and B.

6.the normal to the hyperbola xy=c^2 at the point Q(cq, c/q) intersects thr straight line y=x at R. If O is the origin, show that OQ=QR. if the tangent to the hyperbola at Q intersects OR at P, prove that OP●OR=4c^2.