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Byju's Answer
Standard XII
Mathematics
Discriminant
If the root o...
Question
If the root of the equation
a
x
2
+
b
x
+
c
=
0
are in the ratio
m
:
n
then
A
m
n
a
2
=
(
m
+
n
)
c
2
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B
m
n
b
2
=
(
m
+
n
)
a
c
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C
m
n
b
2
=
(
m
+
n
)
2
a
c
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D
None of these
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Solution
The correct option is
C
m
n
b
2
=
(
m
+
n
)
2
a
c
Let's the roots of the equation
a
x
2
+
b
x
+
c
=
0
are
α
and
β
As the roots are in the ratio
m
:
n
, hence
α
β
=
m
n
Or we can say that
α
=
k
m
and
β
=
k
n
, where
k
is the proportionality constant.
Now the sum of the roots of the equation
α
+
β
=
k
(
m
+
n
)
=
−
b
a
....
(
1
)
and the product of the roots
α
.
β
=
k
2
(
m
n
)
=
c
a
....
(
2
)
By putting the value of
k
from equation
(
1
)
in to equation
(
2
)
, we get,
⇒
b
2
(
m
n
)
(
m
+
n
)
2
a
2
=
c
a
⇒
m
n
b
2
=
(
m
+
n
)
2
a
c
Hence the correct answer is
C
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