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Byju's Answer
Standard X
Mathematics
Nature of Roots
If the equati...
Question
If the equation
(
1
+
m
2
)
x
2
+
2
m
c
x
+
(
c
2
-
a
2
)
=
0
has equal roots, prove that
c
2
=
a
2
(
1
+
m
2
)
.
Open in App
Solution
Given:
(
1
+
m
2
)
x
2
+
2
m
c
x
+
(
c
2
−
a
2
)
=
0
Here,
a
=
(
1
+
m
2
)
,
b
=
2
m
c
and
c
=
(
c
2
−
a
2
)
It is given that the roots of the equation are equal; therefore, we have:
D
=
0
⇒
(
b
2
−
4
a
c
)
=
0
⇒
(
2
m
c
)
2
−
4
×
(
1
+
m
2
)
×
(
c
2
−
a
2
)
=
0
⇒
4
m
2
c
2
−
4
(
c
2
−
a
2
+
m
2
c
2
−
m
2
a
2
)
=
0
⇒
4
m
2
c
2
−
4
c
2
+
4
a
2
−
4
m
2
c
2
+
4
m
2
a
2
=
0
⇒
−
4
c
2
+
4
a
2
+
4
m
2
a
2
=
0
⇒
a
2
+
m
2
a
2
=
c
2
⇒
a
2
(
1
+
m
2
)
=
c
2
⇒
c
2
=
a
2
(
1
+
m
2
)
Hence
proved
.
Suggest Corrections
0
Similar questions
Q.
If roots the equation
x
2
(
1
+
m
2
)
+
2
m
c
x
+
c
2
−
a
2
=
0
are equal, then find
c
.
Q.
If equation has
(
a
2
+
b
2
)
x
2
−
2
(
a
c
+
b
d
)
x
+
(
c
2
+
d
2
)
=
0
equal roots , then prove that
a
b
=
c
d
Q.
(i) If the roots of the equation
a
2
+
b
2
x
2
-
2
a
c
+
b
d
x
+
c
2
+
d
2
=
0
are equal, prove that
a
b
=
c
d
. [CBSE 2017]
(ii) If ad ≠ bc then prove that the equation
a
2
+
b
2
x
2
+
2
a
c
+
b
d
x
+
c
2
+
d
2
=
0
has no real roots. [CBSE 2017]
Q.
If the two circles, which pass through
(
0
,
a
)
and
(
0
,
−
a
)
and touch the line
y
=
m
x
+
c
, will cut orthogonally, if
Q.
If
a
d
≠
b
c
, the prove that the equation
(
a
2
+
b
2
)
x
2
+
2
(
a
c
+
b
d
)
x
+
(
c
2
+
d
2
)
=
0
has no real roots.
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