When you say that the point Q lies on the line PN so that PQ = 2QN, do you mean that Q is between P and N ? Otherwise there are 2 such points
I'm really bad at these types of questions, I don't know what it is about them but they always seem to stump me. Here it is,
The variable point is on the hyperbola with equation and N is the point (3a, 3b). The point Q lies on the line PN so that PQ = 2QN. As t varies find, in cartesian form, the locus of Q.
I tried to find the line PN by finding its gradient by
and then using the usual method for finding the equation of a straight line, that is
but then I get stuck and cannot see what to do next. Do I need to find the point Q? I know I need to eliminate t, but I just can't see how to approach this one.
Are there any general tips and tricks common to conic section questions like this? Given a formula I can identify which conic it is, and its directrix, foci etc. But when the questions get slightly harder (like this one) I just go to pieces.
Any help would be much appreciated
Well as you say we know that
So does this mean I just have to work out the lengths of and and continue the problem that way? I really don't see the way forward and I tend to write out pages of algebra on these types of problems, so I just wanna know I'm going in the right direction
I'm sorry, I have a whole host of conic section problems I can't do. It seems to be a complete brick wall for me. I just can't seem to answer questions on them no matter how much I practice
If have position vectors then the position vector of the point that divides internally in the ratio is given by:
This is a very useful formula. Learn it!
The point P has coordinates ; the point has coordinates , and divides internally in the ratio . (That's what means.) So, using the formula above, the coordinates of are:
So the locus of Q is given by