Parametric Problems (1 Viewer)

[xwrathbringerx]

Hi

1. P(2ap, ap^2) and Q(2aq,aq^2) are 2 points on the parabola x^2 = 4ay. Tangents to the parbaola at P and Q intersect at the point T.

a) Show that the equation of the tangent at P is y=px-ap^2.
b) Find the coordinates of T.
c) P and Q move on the parabola so that the line PQ passes through the point (2a,-a). Show that p + q + 1 = pq.
d) Hence, by finding the Cartesian equation of the locus T, show that T lies on a straight line.
e) With the aid of a diagram, carefully explain why the locus of T is not all of the straight line.


Could someone please show me how to do (e). I've found out that the locus of T is x-y+a = 0 and drawn myself a diagram with all the info provided and obtained but I have no clue how to use these to prove (e).

2. P(2ap, ap^2), Q(2aq,aq^2) and R(2ar,ar^2) are points on the parabola x^2 = 4ay.
a) Show that the equation of the normal at P is py+x = 2ap + ap^3.
b) Find the coordinates of the point of intersection of the normals at P and Q.

Is the pt of intersection (-ap^2q+apq^2 , 2a + a(p^2+pq+q^2)?

c) If the normals P,Q and R are constant, show that p +q +r=0.

How on earth do you do this??

Could someone please help me with these questions? Thanxx

 

[xV1P3R]

e) With the aid of a diagram, carefully explain why the locus of T is not all of the straight line.
Draw lines through point (2a,-a) through which the line PQ has to pass. Notice how there are areas where chords don't exist

One is when the line is completely vertical (not sure if this has any impact) and the other is when P and Q are the same point forming a tangent

From the latter, you work out that T does not exist for x>0.

2b) think it's (-ap^2q - apq^2 , 2a + a(p^2+pq+q^2)

c) For this one, i'm not sure what they mean by constant. But I have a theory that it should say concurrent, not sure.

 

[xwrathbringerx]

Could you please show me what the diagram in (1) looks like because mine looks so confusing and muddled up?

 

[addikaye03]

2. b) Find the coordinates of the point of intersection of the normals at P and Q.

2b)

P: py+x=2ap+ap^3
Q: qy+x=2aq+aq^3

(p-q)y=2a(p-q)+a(p^3-q^3) [Eliminating x]

(p-q)y=2a(p-q)+a(p-q)(p^2+pq+q^2)

y=2a+a[p^2+pq+q^2]

y=a[p^2+pq+q^2+2]

Sub back in to get x.

ap[p^2+pq+q^2+2]+x=2ap+ap^3

ap^2q+apq^2+x=

x=-apq(p+q)

P.O.I : [-apq(p+q) , a(p^2+pq+q^2+2)]

 

[xV1P3R]

Spoiler

Woops, I was wrong about the x>0, now I'm not too sure.

I drew a diagram showing that the chords start when P and Q are the same, and by drawing subsequence chords and tangents, you see the line x-y+a = 0 appearing, but no points for x values past the x value of the tangent

 

[Sunyata]

c) For this one, i'm not sure what they mean by constant. But I have a theory that it should say concurrent, not sure.

Hmmm if it were concurrent, u wuld use the P.O.I. and sub it into the equ. of normal at R rite?

But that doesn't seem to work...you just get this long complicated equation...

The original question with "constant" doesn't seem to help either...

 

[elv09]

for C) i've uploaded my own solution to it ....

http://www.easy-share.com/1909317514/solution to question by e.vuu.pdf