Author Topic: Stormer's Question Thread (Read 616 times) Tweet Share

Stormer · « **on:** January 08, 2010, 07:01:38 pm »

Hey, having a few problems with some anlytic geometry questions, any help would be much appreciated.

1. P(p-1,p^2) and Q(q-1,q^2) are two points on the curve y=(x+1)^2. Find the equation of the chord PQ and hence the equation of the tangent at P. If the tangent at P meets the line x=-1 at the point R, find the equation of the locus of the mid-point of PR.

2. Show that, if the chord joining the points P (ap^2, 2ap), Q(aq^2, 2aq) on the parabola y^2=4ax passes through (a,0) then pq=-1. Further, the tangent at P meets the line through Q parallel to the axis of the parabola at R. Prove that the line x+1=0 bisects PR.

kamil9876 · « **Reply #1 on:** January 08, 2010, 07:43:04 pm »

1.) $\frac{y-p^2}{x-p+1}=\frac{q^2-p^2}{q-p}$

$y=\frac{q^2-p^2}{q-p}(x-p+1)+p^2$

$y=(q+p)(x-p+1)+p^2$ (equation of chord)

letting $q \rightarrow p$ we get the equation of the tangent:

$y=(2p)(x-p+1)+p^2$

$R=(-1,-2p^2+p^2)=(-1,-p^2)$ (subbing in x=-1)

Therefore the midpoint is: $(\frac{(p-1)+(-1)}{2} , \frac{p^2+(-p^2)}{2})$

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