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Other
Quantitative Aptitude
Quadratic Equations
If the equati...
Question
If the equation
(
1
+
m
2
)
x
2
+
2
m
c
x
+
c
2
−
a
2
=
0
has equal root, show that
c
2
=
a
2
(
1
+
m
2
)
Open in App
Solution
(
1
+
m
2
)
x
2
+
2
m
c
x
+
c
2
−
a
2
=
0
For the roots to be equal, discriminant =
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
−
4
a
2
+
4
m
2
c
2
−
4
m
2
a
2
4
a
2
+
4
m
2
a
2
=
4
c
2
a
2
(
1
+
m
2
)
=
c
2
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1
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 the equation
(
a
2
+
b
2
)
x
2
−
2
(
a
c
+
b
d
)
x
+
c
2
+
d
2
=
0
has equal roots, then
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 each pair of equations
a
1
x
2
+
b
1
x
+
c
1
=
0
,
a
2
x
2
+
b
2
x
+
c
2
=
0
and
a
3
x
2
+
b
3
x
+
c
3
=
0
has a common root, then show that
(i)
c
1
a
2
+
c
2
a
1
c
1
a
2
−
c
2
a
1
+
a
1
a
2
b
3
a
3
(
a
1
b
2
−
a
2
b
1
)
=
0
(ii)
(
a
1
b
2
−
a
2
b
1
a
1
c
2
−
a
2
c
1
)
2
=
a
1
a
2
c
3
c
1
c
2
a
3
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