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Mathematics
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© examsnet.com
Question : 93
Total: 110
If tangents are drawn to the ellipse
x
2
+
2
y
2
= 2 at all points on the ellipse otherthan its four vertices then the mid points of thetangents intercepted between the coordinateaxes lie on the curve :
[11 Jan 2019 Shift 1]
x
2
2
+
y
2
4
= 1
x
2
4
+
y
2
2
= 1
x
2
2
x
2
+
y
2
4
y
2
= 1
x
2
4
x
2
+
y
2
2
y
2
= 1
Validate
Solution:
Equation of general tangent on ellipse
x
a
s
e
c
θ
+
y
b
c
o
s
e
c
θ
= 1
a =
√
2
, b = 1
⇒
x
√
2
s
e
c
θ
+
y
c
o
s
e
c
θ
= 1
Let the midpoint be (h, k)
h =
√
2
s
e
c
θ
2
⇒ cos θ =
1
√
2
h
and k =
c
o
s
e
c
θ
2
⇒ sin θ =
1
2
k
Since
s
i
n
2
θ
+
c
o
s
2
θ
= 1
⇒
1
2
h
2
+
1
4
k
2
= 1
1
2
x
2
+
1
4
y
2
= 1
© examsnet.com
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