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Date May 2018 Marks available 2 Reference code 18M.2.hl.TZ0.5
Level HL only Paper 2 Time zone TZ0
Command term Determine Question number 5 Adapted from N/A

Question

Consider the ellipse x2a2+y2b2=1.

The area enclosed by the ellipse is 8π and b=2.

Show that the area enclosed by the ellipse is πab.

[9]
a.

Determine which coordinate axis the major axis of the ellipse lies along.

[2]
b.i.

Hence find the eccentricity.

[2]
b.ii.

Find the coordinates of the foci.

[2]
b.iii.

Find the equations of the directrices.

[2]
b.iv.

The centre of another ellipse is now given as the point (2, 1). The minor and major axes are of lengths 3 and 5 and are parallel to the x and y axes respectively. Find the equation of the ellipse.

[3]
c.

Markscheme

A=4ydx      (M1)

x2a2+y2b2=1

y=ba2x2a      (A1)

let x=acosθy=bsinθ      M1

dxdθ=asinθ      A1

when x=0,θ=π2. When x=a,θ=0       A1

A=40π2bsinθ(asinθ)dθ      M1

A=4ab0π2sin2θdθ

A=2ab0π2(1cos2θ)dθ      M1

A=2ab[θsin2θ2]0π2     A1

A=2ab[00(π20)]     M1

A=πab      AG

[9 marks]

a.

b=2

hence 2πa=8πa=4      A1

hence major axis lies along the x-axis      A1

[2 marks]

b.i.

b2=a2(1e2)      (M1)

4=16(1e2)e=32      A1

[2 marks]

b.ii.

coordinates of foci are (±ae,0)=(23,0),(23,0)      A1A1

[2 marks]

b.iii.

equations of directrices are x=±ae=83,83      A1A1

[2 marks]

b.iv.

a=32,b=52     (A1)

hence equation is 49(x2)2+425(y1)2=1     M1A1

[3 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.
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b.iii.
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b.iv.
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c.

Syllabus sections

Topic 2 - Geometry » 2.6 » Conic sections.

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