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Byju's Answer
Standard XII
Mathematics
Distance Formula
The number of...
Question
The number of tangents to the circle
x
2
+
y
2
=
3
that are normals to the ellipse
x
2
9
+
y
2
4
is
A
one
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B
two
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C
three
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D
zero
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Solution
The correct option is
A
one
Let y = mx+c is tangent to
x
2
+
y
2
=
3
Then by condition of tangeney
∣
∣
∣
c
√
m
2
+
1
∣
∣
∣
=
√
3
⇒
c
2
=
3
(
m
2
+
1
)
__(1)
y
=
m
x
+
c
is Normal to
x
2
9
+
y
2
4
=
1
If
c
=
(
b
2
−
a
2
)
m
√
a
2
+
b
2
m
2
⇒
c
=
(
4
−
9
)
m
√
9
+
4
m
2
⇒
c
2
=
25
m
2
4
m
2
+
9
⇒
3
(
m
2
+
1
)
(
4
m
2
+
9
)
=
25
m
2
let
m
2
=
t
⇒
3
(
t
+
1
)
(
4
t
+
9
)
=
25
t
since,
D
<
0.
Hence, There is no root :
Hence, There is no tangent to circle which is
Normal to ellipse.
Suggest Corrections
0
Similar questions
Q.
How many tangents to the circle
x
2
+
y
2
=
3
are there which are normal to the ellipse
x
2
9
+
y
2
4
=
1
Q.
The slopes of the common tangents of the ellipse
x
2
9
+
y
2
2
=
1
and the circle
x
2
+
y
2
=
3
are
Q.
If tangents are drawn from any point on the circle
x
2
+
y
2
=
25
to the ellipse
x
2
16
+
y
2
9
=
1
,
then the angle between the tangents is
Q.
Equation of the common tangent to the ellipse
x
2
16
+
y
2
9
=
1
and the circle
x
2
+
y
2
=
12
is
Q.
Assertion :Tangents drawn from any point on the circle
x
2
+
y
2
=
225
to the ellipse
x
2
144
+
y
2
81
=
1
are at right angle. Reason: Equation of the auxiliary circle of the ellipse
x
2
a
2
+
y
2
b
2
=
1
is
x
2
+
y
2
=
a
2
.
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