# P5 Ellipses

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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

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

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#2

(Original post by

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

**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

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

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#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.

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.

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#5

(Original post by

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.

**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.

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.

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#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.

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