Parametrics Problem (1 Viewer)

[EpikHigh]

Hey guys kinda stuck on this question ending up in deadends.

11) P is a variable point on the parabola x^2 = -4y. The tangent from P cuts the parabola x^2 = 4y at Q and R. show that 3x^2 = 4y is the equation of the locus of the mid-point of the chord RQ. Question is from exercise 24(c) Fitzpatrick 3U Y11/12

 

[braintic]

Hey guys kinda stuck on this question ending up in deadends.

11) P is a variable point on the parabola x^2 = -4y. The tangent from P cuts the parabola x^2 = 4y at Q and R. show that 3x^2 = 4y is the equation of the locus of the mid-point of the chord RQ. Question is from exercise 24(c) Fitzpatrick 3U Y11/12

Let Q and R have standard parameters p & q.
Equation of chord QR is y=(p+q)/2 . x - pq (you derive)
Solve this simultaneously with the parabola x^2 = -4y, and set the discriminant to zero (its a tangent, so only one point of intersection).
You should get (p+q)^2 = -pq.
Midpoint of PQ has coords [p+q, (p^2 + q^2)/2 ].
See if you can finish it off from here.

 

[Study notes]

The way I did it took awhile but: Using P(2p , -p^2) on the parabola x^2 = -4y:

dy/dx = x^2/-4
=2x / -4
m = 2(2p) / -4
= -p

Get the equation of the tangent which is:

y = -px +p^2 ..... (1)
y = x^2 / 4 ..... (2) (this is just x^2 = 4y rearranged)

Solve it Simultaneously to get:

x^2 = -4px + 4p^2
x^2 + 4px - 4p^2 = 0

Now, using the discriminate:

Delta = (4p)^2 - 4(1)(-4p^2)
= 36p^2

Put that into the quadratic equation of:

x = -4p +or- sqroot(36p^2)
_____________________
2

so you get two x values, one + and the other is - .

The rest is fairly straight forward, give it go from here

(sorry if I made any calculation errors, basically I used discriminates to find x, then y etc.)

 

[EpikHigh]

fml I'm still not getting 3x^2 = 4y wtf is wrong with me

 

[Study notes]

Tell me what you get as the x and y values of point Q and R

 

[braintic]

fml I'm still not getting 3x^2 = 4y wtf is wrong with me

In case you're doing it my way, I made a mistake below. You should get (p+q)^2 = -4pq.
Does that make a difference?

 

[Study notes]

you can go down this path too:

skipped to: x^2 + 4px - 4p^2 = 0 (scroll up to check how I got here)

using the sum of roots (Alpha + Beta):

A + B = -b/a
= -4p / 1
= -4p
This means that one of the roots plus the other one of the roots equal to -4p. So, the roots = x, therefore to find the mid point of the two roots (x value)

-4p / 2 = x
x = -2p

go from there to get y and then work out the locus