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.
Determine which coordinate axis the major axis of the ellipse lies along.
Hence find the eccentricity.
Find the coordinates of the foci.
Find the equations of the directrices.
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.
Markscheme
A=4∫ydx (M1)
x2a2+y2b2=1⇒
y=b√a2−x2a (A1)
let x=acosθ⇒y=bsinθ M1
dxdθ=−asinθ A1
when x=0,θ=π2. When x=a,θ=0 A1
⇒A=4∫0π2bsinθ(−asinθ)dθ M1
⇒A=−4ab∫0π2sin2θdθ
⇒A=−2ab∫0π2(1−cos2θ)dθ M1
⇒A=−2ab[θ−sin2θ2]0π2 A1
⇒A=−2ab[0−0−(π2−0)] M1
⇒A=πab AG
[9 marks]
b=2
hence 2πa=8π⇒a=4 A1
hence major axis lies along the x-axis A1
[2 marks]
b2=a2(1−e2) (M1)
4=16(1−e2)⇒e=√32 A1
[2 marks]
coordinates of foci are (±ae,0)=(2√3,0),(−2√3,0) A1A1
[2 marks]
equations of directrices are x=±ae=8√3,−8√3 A1A1
[2 marks]
a=32,b=52 (A1)
hence equation is 49(x−2)2+425(y−1)2=1 M1A1
[3 marks]