parametric help plz (1 Viewer)

[Mehae]

Points P(10p, 5p^2) and Q(10q, 5q^2) lie on the parabola x^2=20y. find the equation of the locus of the midpoint of PQ if pq =-2

plz and thank you

 

[SeCKSiiMiNh]

use ur midpoint formula first to find midpoint of pq:

u shld get x = (10p + 10q) / 2
and y = (5p^2 + 5q^2) / 2

simplify that and you'll get:
1 - x = 5(p+q)
2 - y=2.5 (p^2 + q^2)

note that p^2 + q^2 = (p+q)^2 - 2pq

from 1, p+q = x/5

so sub that into y, and u have a value for pq so sub that in too

done

 

[Mehae]

im still getting the wrong answer

the book says x^2-10y+100

 

[Mehae]

its a random sheet, but ill check it up with my teacher. thanks for the help

 

[Luxxey]

Okay so your midpoint is:
x: 5(p+q)
y: 2.5(p^2 + q^2)

Square both sides of your x co-ord:
x^2 = 25[(p+q)^2]
(p+q)^2 = x^2/25

Now:
y = 2.5(p^2 + q^2)
= 2.5[(p+q)^2 - 2pq]

Using (p+q)^2 = x^2/25 and pq=-2,

y = 2.5[(x^2/25) + 4]
y = x^2/10 + 10

Multiply both sides by 10 and re-arrange:

x^2 - 10y + 100 = 0.

Which is your solution.