Parametrics Q (1 Viewer)

[shaon0]

P(2ap,ap^2) and Q(2aq,aq^2) are two points on the parabola x^2=4ay. The tangents TP and TQ drawn from an external point T cross the x-axis at A and B respectively.
i) Find the co-ordinates of A and B.
I know that A and B are A(x1,0) and B(x2,0) but i don't know what to do next.

 

[Dr. Zoidberg]

 

[shaon0]

Some help needed...
ii) Find the equation of the line passing through the focus that bisects the line AB.
iii) Hence, show that this line is perpendicular to the chord of contact T.

 

[lyounamu]

Some help needed...
ii) Find the equation of the line passing through the focus that bisects the line AB.
iii) Hence, show that this line is perpendicular to the chord of contact T.

question?

 

[shaon0]

P(2ap,ap^2) and Q(2aq,aq^2) are two points on the parabola x^2=4ay. The tangents TP and TQ drawn from an external point T cross the x-axis at A and B respectively.
i) Find the co-ordinates of A and B.
I know that A and B are A(x1,0) and B(x2,0) but i don't know what to do next.

the one above

 

[lolokay]

focus is at (0,a)
AB is (ap, 0) , (aq, 0)
it's midpoint is (a(p+q)/2 , 0)

gradient of line through focus and midAB is -a/[a(p+q)/2] = -2/(p+q)
y-a = -2x/(p+q)
2x + y(p+q) - a(p+q) = 0

chord of contact is the line joining P(2ap,ap^2) and Q(2aq,aq^2).
it's gradient is a(p+q)(p-q)/2a(p-q) = (p+q)/2
(p+q)/2 * -2/(p+q) = -1

therefore they are perpendicular

 

[-tal-]

focus is at (0,a)
AB is (ap, 0) , (aq, 0)
it's midpoint is (a(p+q)/2 , 0)

gradient of line through focus and midAB is -a/[a(p+q)/2] = -2/(p+q)
y-a = -2x/(p+q)
2x + y(p+q) - a(p+q) = 0

what's a chord of contact?

A chord of contact is the line in between the parabola that touches points Q & P.

 

[Continuum]

P: y=px-ap²
Q: y=qx-aq²
px-ap²=qx-aq²
Rearrange to get: x=a(p+q)
Sub in x into one of the equations to get: y=-apq
T: a(p+q),-apq
Chord of contact: xx₀=2a(y+y₀)
ax(p+q)=2a(y-apq)
Rearrange to form y=(p+q)/2 . x + apq
Gradient of chord of contact: (p+q)/2
(p+q)/2 . -2/(p+q) = -1
Therefore perpendicular

 

[shaon0]

thanks for the help guys. Really appreciate it.

 

[shaon0]

1. If the chord of contact to the parabola x^2=4ay from an external point P(x0,y0) passes through the point A(0,2a), show that the locus of the midpoint of PA is the x-axis.
.....I have the midpoint of PA, what should i do next?
2.The tangent at the point P(2ap,ap^2) on the parabola x^2=4ay cuts the y-axis at T. The line via the focus S parallel to this tangent cuts the directrix at V. M is the midpoint of TV. Find the locus of M as P moves on the parabola.

 

[NitroBoon]

To determine the locus of a parametric point, all you have to do is to eliminate the parameter(s) from the coordinate

 

[lolokay]

in 1. you should get pq = -2, and y0 = apq = -2a, so the midpoint of 2a and -2a is the x axis

 

[NitroBoon]

2.The tangent at the point P(2ap,ap^2) on the parabola x^2=4ay cuts the y-axis at T. The line via the focus S parallel to this tangent cuts the directrix at V. M is the midpoint of TV. Find the locus of M as P moves on the parabola.

Solution:

<o></o>Feel free to post corrections, I am not too sure about the answer lol, seems abit random.

 

[shaon0]

Solution:

<o></o>Feel free to post corrections, I am not too sure about the answer lol, seems abit random.

thanks for the solytion. i got most of the question write in the beginning but i made a stupid mistake on expanding and getting the Locus of M.
Thanks

 

[Aerath]

How do you do a solution on the computer like that?

 

[AnandDNA]

How do you do a solution on the computer like that?

Go to school

 

[Dr. Zoidberg]

How do you do a solution on the computer like that?

Word 2007. Insert equation.

 

[kaz1]

How do you do a solution on the computer like that?

I think he might be using Microsoft Maths.