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Title: Cambridge international AS and A level mathematics 9709
Description: To find equation of line intersecting an ellipse and the skills related to the area of an ellipse. Figure are added to make concept easier.
Description: To find equation of line intersecting an ellipse and the skills related to the area of an ellipse. Figure are added to make concept easier.
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geometry
shahbaz ahmed
January 2024
(a)
The diagram shows a cure with equation
x2
a2
+
y2
b2
= 1
...
(i)Find the equation of the line that passes through A and B
...
Give your answer as a multiple of π
(ii)
A curve mathematically similar to the one in diagram ,intersect the x=axis at (12,0) and (-12,0)
Work out the area enclosed by this curve ,giving your answer as a multiple of π
...
4
a2
+
0
b2
=1
16
a2
+0=1
16
a2
=1
x2
a2
2
=⇒
a2 = 16
Again ,putting (0, 2) in the equation
02
a2
+
22
b2
= 1
...
0+
4
b2
=1
b2 = 4
(b)
Putting P(2,k) and Q(2,-k) in the equation
x2
16
+
y2
4
=1
22
k2
+
22
k2
=
22
k2
+
22
(−k)2
=1
Same equation at points P(2,k) and Q(2,-k)
3
4
k2
+
4
k2
=1
2( k42 ) = 1
8
k2
=1
k2 = 8
√
√
k = ± 8 = ±2 2
(ii)
Now to Calculate angle POQ
√
Slope of the line passing through O(0, 0) and P(2, 8)
m1 =
√
8
2
√
Slope of the line passing through O(0, 0) and Q(2, − 8)
m2 =
√
− 8
2
Angle between the line intersecting at point O will be:
m2 −m1
angleP OQ = arctan 1+m
1 m2
√
√
8
8
2 √− 2√
− 8
1+[ 2 ][ 28 ]
√
−2( 28 )
√
1−[ 28 ]2
√
−
̸
P OQ = arctan
̸
P OQ = arctan
̸
− 8
P OQ = arctan 1−[
8
]
̸
P OQ = arctan −1−28
̸
P OQ = arctan −−18
√
′
P OQ = arctan 8 = 70◦ 31 43
...
=1
x2 = a2
Putting x = 12
a2 = 122
a = 12, a = −12
Similarly Putting x = 0 in the equation
y2
b2
x2
a2
+
y2
b2
= 1
...
y 2 = 122
Putting x = 0, y 2 = 122 in the equation
x2
a2
+
2
12
b2
y2
b2
= 1
...
b2 = 122
b = 12, b = −12
Since
Area = πab
5
=⇒
Area = π(12)(12) = 144π
6
Title: Cambridge international AS and A level mathematics 9709
Description: To find equation of line intersecting an ellipse and the skills related to the area of an ellipse. Figure are added to make concept easier.
Description: To find equation of line intersecting an ellipse and the skills related to the area of an ellipse. Figure are added to make concept easier.