Equations of Ellipse Conics HSC Maths Extension 2

Size: px

Start display at page:

Download "Equations of Ellipse Conics HSC Maths Extension 2"

Alexandra Merritt
5 years ago
Views:

1 Equations of Ellipse HSC Maths Extension

2 1 Question 1 Find the equation of the ellipse whose foci on the y-axis, centre 0,0, a, b. Question Find the equation of the ellipse whose foci 4,0, b. Question Find the equation of the ellipse whose foci 0,, a. Question 4 Find the equation of the ellipse with foci at 6,0 and eccentricity of. Question Find the equation of the ellipse with foci at,0 and major axis of 1 units. Question 6 Find the equation of the ellipse with foci at,0 and major axis of 10 units. Question 7 y mx 8 is a chord of the ellipse x 4y 16 that passes through 0,8 for all real values of m. Find the locus of the midpoint of the chord as m varies. Question 8 Sketch 1, showing the foci and the directrices. 9 4

3 Equations of Ellipse Question 9 Sketch 1, showing the foci and the directrices Question 10 Sketch x 4y 144, showing the foci and the directrices. Question 11 Sketch 1, showing the foci and the directrices. 4 Question 1 Sketch x 1 y 1, showing the foci and the directrices Question 1 Sketch x1 y 1, showing the foci and the directrices. 8 4 Question 14 Sketch 1, showing the foci and the directrices. 4 9 Question 1 y Sketch x 1, showing the foci and the directrices. 4

4 Question 16 (a) Prove that the equation of the chord joining Pacos, bsin and Qacos, bsin of the ellipse x a y is abcos bx cos ay sin. b 1 (b) Hence, obtain the equation of the tangent to the ellipse at the point Pacos, bsin. (c) If PQ is a focal chord, show that cos ecos. Question 17 (a) Explain why the parametric equations of the ellipse are cos,sin 4x 9y 6. (b) What are the parametric x1 y 1? 4 Question 18 Write the equation of the locus of a point P that moves such that its distance from,1 is half as its distance from the line 0. Question 19 Write the equation of the locus of a point that moves so that its distance from,1 is half as its distance from the line y x.

5 4 Equations of Ellipse Fully Worked Solutions Question Question 1 9 Question Question 4 Since ae 6 and e, a 9 b a 1 e gives b 4 Put in Question 1 1 Since ae and a 6, e b a 1 e gives b 7 Put in Question 6 10 ae and a, e b a 1 e gives b Question 7 Solve both equations simultaneously, x mx 1 4m x 64mx m x 64mx 40 0 Sum of roots b m The midpoint x a 1 4m m 8 y mx m 14m x x 4m m 4 y 4y Put to (4) x 4y y 8 y y x 4y x y

Question 8 4 9 1e 4 e 1 9 9 e Foci,0, Directrices x 9

6 Question e 4 e e Foci,0, Directrices x 9 Question e 4 e e Foci,0, Directrices x 8

6 Equations of Ellipse Question 10 6 144 1e 6 e 1 144 4 e Foci 6,0,

7 6 Equations of Ellipse Question e 6 e e Foci 6,0, Directrices 4 x 8 Question e 4 1 e 1 1 e Foci 1,0, Directrices x

7 Question 1 This ellipse has centre 1,0.

8 7 Question 1 This ellipse has centre 1, e e 1, therefore e Foci 1 7,0, Directrices x 1 7 Question 1 This ellipse has centre Foci 1,, and 1, Directrices x 1 4 x or x 1, e, 4 1 e 1, therefore 8 e 1

9 8 Equations of Ellipse Question 14 4 e 1, therefore 9 9 Foci 0,, Directrices e y 9 Question 1 1 e e Foci 0,, Directrices y 4

10 9 Question 16 sin x y sinxcos y sinycos x 1 (a) sin x y sinxcos y sinycos x 1 gives sin x y sin x y sinycos x let xy and xy, then x and y becomes sin sin sin cos 4 cos x y cos xcos y sinx siny x y cos cos cos sin sin 6 6 gives cos x y cos x y sinxsiny, cos cos sin sin sin sin gradient of PQ b b a cos a cos b equation of sin bsin PQ: y bsin x acos acos acos a y bsin cos cos b sin sin x acos ay bsin sin sin bsin cos x acos ay bsinsin bcos x acos abcos cos absin sin xbcos aysin ab cos cos sin sin xbcos aysin abcos xbcos ay sin abcos bx cos ay sin (b) Let, the equation of PQ becomes bcosx asiny ab (c) Substituting ae,0 into the equation of PQ, bcos ae abcos ecos cos The positive sign if PQ passes through the positive focus. Question 17 (a) Substituting x cos and y sin into 4x 9y 6 LHS 4 cos 9 sin 6 cos sin 6 RHS (b) Let x 1 cos, y sin x 1 cos, y sin Question 18

11 10 Equations of Ellipse Let P and S be xy, and,1 respectively. Also, let M be the foot of the perpendicular from P onto the directrix x y 0 1 SP x y xy PM 1 SP PM 4SP PM x y xy 4 1 8x x 8y 16y 8 x y 4 xy 4x 4y 7x 7y 8x 1y xy 6 0 Question 19 1 SP x y x y PM 1 SP PM 4SP PM x y x y The equation of the ellipse is 7x 7y x 16y xy 40 0

12 11

Edexcel GCE A Level Maths. Further Maths 3 Coordinate Systems

Edexcel GCE A Level Maths. Further Maths 3 Coordinate Systems Edecel GCE A Level Maths Further Maths 3 Coordinate Sstems Edited b: K V Kumaran kumarmaths.weebl.com 1 kumarmaths.weebl.com kumarmaths.weebl.com 3 kumarmaths.weebl.com 4 kumarmaths.weebl.com 5 1. An ellipse