You are Here: Home >< Maths

# P5 Ellipses watch

1. Its q.16 p.88 edexcel p5

show that S(root3,0) is a focus of the ellipse eq/n 3x^2 + 4y^2 = 36. O is origin, P is a pt. on ellipse, a line is draw from O perp. to the tangent to the ellipse at P and this line meets the line SP (produced if neccessary) at the pt. Q. Show that the locus of Q is a circle.

can do the proof bit but not the locus job, help wud be good than q
2. (Original post by Gregball_87)
Its q.16 p.88 edexcel p5

show that S(root3,0) is a focus of the ellipse eq/n 3x^2 + 4y^2 = 36. O is origin, P is a pt. on ellipse, a line is draw from O perp. to the tangent to the ellipse at P and this line meets the line SP (produced if neccessary) at the pt. Q. Show that the locus of Q is a circle.

can do the proof bit but not the locus job, help wud be good than q
ill have a go. sorry if it contans errors too messy doing on machine.
let P=(x1,y1)
from eqn of ellipse 3x+4ydy/dx=0
dy/dx=-3x1/4y1 at P
line perp to tangent at P has grad 4y1/3x1
since this line passes through (0,0) eqn of line OT,say, is y=4y1x/3x1
eqn of SP is given by
y=y1(x-rt(3))/(x1-rt(3))
the two lines meet when
4y1x/3x1=y1(x-rt(3))/(x1-rt(3))
(x1-4rt(3))x=-3rt3x1
ie x=-3rt(3)x1/(x1-4rt(3)) ....................(1)
this gives y=-4rt(3)y1/(x1-4rt(3))..........(2)
(1) gives x1x-4rt(3)x+3rt(3)x1=0
x1(x+3rt(3))=4rt(3)x
x1=4rt(3)x/(x+3rt(3)).....................( 3)
using (3) gives
x1-4rt(3)=4rt(3)x/(x+3rt(3))-4rt(3)=(4rt(3)x-4rt(3)x-36)/(x+3rt(3))
=-36/(x+3rt(3))...................... .(4)
putting (4) into (2) gives
y=rt(3)y1(x+3rt(3))/9
so 9y/(rt(3)(x+3rt(3))=y1............. ..........(5)
since x1 y1 lie on ellipse they satisfy eqn of ellipse this gives
3.16.3x^2/(x+3rt(3))^2+81.4y^2/3(x+3rt(3))^2=36
144x^2+108y^2=36(x+3rt(3))^2
=36x^2+6.36rt(3)x+27.36
108x^2-216rt(3)x+108y^2=972
x^2-2rt(3)+y^2=9
(x-rt(3))^2+y^2=12 well theres hope, that is eqn of circle
3. jesus wept, no wonder i struggled

thanks alot
4. This question is about ellipses. I'd be grateful if someone could shed some light on where my solution went wrong:

An ellipse has focus S (rt5, 0) and equation (x^2)/9 + (y^2)/4 = 1
The variable point T(3cost, 2sint) is joined to S. The line ST is produced to P so ST/SP = (1/3). Find the locus of P as t varies.

Firstly i rearranged the ratio expression to give 3ST=SP.
First i tried to find a parametric expression for ST.
ST: x=3cost-rt5. y=2sint (as ST = T - S?)
SP: x=9cost-3rt5. y=6sint
cost = (x+3rt5)/9. sint=y/6
(x+3rt5)^2 /81 + y^2/36 = 1
4(x+3rt5)^2 + 9y^2 = 324.
However, the book give the same answer but the coefficient of rt5 is 2.
I'd greatly appreciate any help.
5. (Original post by Gaz031)
This question is about ellipses. I'd be grateful if someone could shed some light on where my solution went wrong:

An ellipse has focus S (rt5, 0) and equation (x^2)/9 + (y^2)/4 = 1
The variable point T(3cost, 2sint) is joined to S. The line ST is produced to P so ST/SP = (1/3). Find the locus of P as t varies.

Firstly i rearranged the ratio expression to give 3ST=SP.
First i tried to find a parametric expression for ST.
ST: x=3cost-rt5. y=2sint (as ST = T - S?)
SP: x=9cost-3rt5. y=6sint
cost = (x+3rt5)/9. sint=y/6
(x+3rt5)^2 /81 + y^2/36 = 1
4(x+3rt5)^2 + 9y^2 = 324.
However, the book give the same answer but the coefficient of rt5 is 2.
I'd greatly appreciate any help.
x cord of t is 3cost so x-cord of p is 9cost-2rt(5)
eg if S was at (2,0) and P at (5,4) then SP=rt(9+16)=5 so we would need ST to have length 15 so T would have coords (11,12) so ST has length rt(81+144)=15.
ie x-cord of T is 3xcord of P- 2.2
y cord of T is 6sint
x=9cost-2rt(5)
gives 2(x+2rt(5))=18cost
3y=18sint
so 4(x+2rt(5))^2+9y^2=324.
hope this ok in a rush in a lecture.
6. Thanks. I understand now. It helps me to think of it more from the vector side of things. Ie: First find the change from S to T, then from S to T, finally add the position vector of S to get the final coordinates.

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: February 4, 2005
Today on TSR

### TSR Pub Quiz 2018 - Anime

The first of our week-long series of entertainment quizzes

### University open days

Wed, 21 Nov '18
• Buckinghamshire New University
Wed, 21 Nov '18
• Heriot-Watt University
Wed, 21 Nov '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams