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Joint Entrance Examination
Mathematics
Ellipse
Question 19
Question 19 of 183
Medium
MCQ
Let
C
C
C
be the circle of minimum area enclosing the ellipse
E
:
x
2
a
2
+
y
2
b
2
=
1
E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1
E
:
a
2
x
2
+
b
2
y
2
=
1
with eccentricity
1
2
\frac{1}{2}
2
1
and foci
(
±
2
,
0
)
( \pm 2,0)
(
±
2
,
0
)
. Let
P
Q
R
P Q R
P
QR
be a variable triangle, whose vertex
P
P
P
is on the circle
C
C
C
and the side...
A
8
(
3
+
2
)
8(3+\sqrt{\sqrt{2}})
8
(
3
+
2
)
B
8
(
2
+
3
)
8(2+\sqrt{\sqrt{3}})
8
(
2
+
3
)
C
6
(
3
+
2
)
6(3+\sqrt{\sqrt{2}})
6
(
3
+
2
)
D
6
(
2
+
3
)
6(2+\sqrt{\sqrt{3}})
6
(
2
+
3
)
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Official solution
=
1
2
×
(
a
+
b
)
⋅
2
a
=
a
(
a
+
b
)
e
=
1
2
a
e
=
2
⇒
a
=
4
,
b
=
2
3
⇒
=
4
(
4
+
2
3
)
=
8
(
2
+
3
)
\begin{gathered} =\frac{1}{2}\times (a+b)\cdot 2a=a(a+b) \\ e=\frac{1}{2}ae=2\Rightarrow a=4,b=2\sqrt{3} \\ \Rightarrow =4(4+2\sqrt{3})=8(2+\sqrt{3}) \end{gathered}
=
2
1
×
(
a
+
b
)
⋅
2
a
=
a
(
a
+
b
)
e
=
2
1
a
e
=
2
⇒
a
=
4
,
b
=
2
3
⇒=
4
(
4
+
2
3
)
=
8
(
2
+
3
)
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