Parametric Equations of the Parabola (2 Viewers)

[FDownes]

Thanks. Now time for another;

How can I solve the two equations x + py = p³ + 2p and x + qy = q³ + 2q simultaneously to find the point of intersection (-pq(p + q), p² + q² + pq + 2)?

EDIT: Never mind, got it. Thanks again Tommy.

 

[FDownes]

I'd like to double check my working for this question to make sure I have it right;

a) Write down the equation of the chord joining the points P (2ap, ap²) and q (2aq, aq²) on the parabola x² = 4ay.

[I think I've got this working down. I'm not sure if it should plus apq or minus apq in the final equation though...]

= m = [(ap²) - (aq²)]/[(2ap) - (2aq)]
= m = [a(p² - q²)]/[2a(p - q)]
= m = [a(p - q)(p + q)]/[2a(p - q)]
= m = (p + q)/2
= [y - (ap²)] = [(p + q)/2][x - (2ap)]
= y - ap² = [x(p + q)/2] - [2ap(p + q)/2]
= y - ap² = 1/2(p + q)x - ap² - apq
= y = 1/2(p + q)x - apq

b) Show that if PQ is a focal chord then pq = -1

= focus = (0, 1)
= 1 = 1/2(p + q)(0) - (1)pq
= -pq = 1
= pq = -1

c) Find the equation of the tangent at P and the coordinates of T, the point of intersection of the tangents at P and Q. Hence determine the equation of the locus of T as P and Q vary.

[For this question and the one below I have some attempted working, but it's fairly messed up. I'd like to check it by comparing it to someone elses working, if they could post it here. The tangent for P should be y - px + ap² = 0, and likewise the tangent for Q should be y - qx + aq² = 0. The coordinates of T are
[a(p + q), apq].]

d) Find the equation of the normal at P and the coordinates of N, the point of intersection of the normals of P and Q. Hence determine the equation of the locus of N.

 

[tommykins]

Attached solutions to them.

http://img168.imageshack.us/img168/8523/67778570bz2.jpg

Your question b) is done incorrectly, you need to be subbing in y = a since the focii is (0,a) not (0,1).

 

[FDownes]

Hmm... I followed a similar process when trying to figure out c), but couldn't get the correct answer. According to the sheet, the locus of T should be y = -a (i.e. the directrix).

While I'm here I may as well post the the answers to d) as well. N should be [-apq(p + q), a(p² + pq + q² + 2)], and have a locus of x² = a(y - 3a). This is from the sheet, not from my own working.