parametrics question (1 Viewer)

[FD3S-R]

19. P(2ap, ap^2) and Q(2aq, aq^2) are points on the parabola x^2=4ay/ The Tangents at P and Q meet at R, and R lies on the parabola x^2=-4ay.

(a) Show R coordinates (a(p+q), apq)
(b) Show that p^2 + q^2 +6pq = 0
(c) As P and Q vary, show that the locus of the midpoint of the chord PQ is the parabola 3x^2 = 4ay


i can do a and b but i cant get rid of pq in c, it keeps repeating

 

[shafqat]

19. P(2ap, ap^2) and Q(2aq, aq^2) are points on the parabola x^2=4ay/ The Tangents at P and Q meet at R, and R lies on the parabola x^2=-4ay.

(a) Show R coordinates (a(p+q), apq)
(b) Show that p^2 + q^2 +6pq = 0
(c) As P and Q vary, show that the locus of the midpoint of the chord PQ is the parabola 3x^2 = 4ay


i can do a and b but i cant get rid of pq in c, it keeps repeating

for the locus,
y = a(p^2 + q^2)/2
x = a(p+q)

now 3x^2 = 3a(p^2 + q^2 + 2pq)
from the result in b, 2pq = -(p^2 + q^2)/3
so 3x^2 = 3a(p^2 + q^2-(p^2 + q^2)/3)
= 2/3*3*a(p^2 + q^2)
= 2a(p^2 + q^2)
= 4ay

 

[Jago]

doesn't 3x^2 = 3a^2(p+q)^2?

 

[shafqat]

doesn't 3x^2 = 3a^2(p+q)^2?

yep, sorry abt that
it still works, just replace the a in the working with a^2