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Joint Entrance Examination
Mathematics
Quadratic Equations and Inequalities
Question 81
Question 81 of 157
Medium
MCQ
If the sum of the roots of the quadratic equation
a
x
2
+
b
x
+
c
=
0
a{x^2} + bx + c = 0
a
x
2
+
b
x
+
c
=
0
is equal to the sum of the squares of their reciprocals, then \[{a \over c},\,{b \over a}\] and \[{c \over b}\] are in
A
Arithmetic - Geometric Progression
B
Arithmetic Progression
C
Geometric Progression
D
Harmonic Progression
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Official solution
a
x
2
+
b
x
+
c
=
0
,
a{x^2} + bx + c = 0,
a
x
2
+
b
x
+
c
=
0
,
\[\alpha + \beta = {{ - b} \over a},\alpha \beta = {c \over a}\] As for given condition, \[\alpha + \beta = {1 \over {{\alpha ^2}}} + {1 \over {{\beta ^2}}}\] \[\alpha + \beta = {{{\alpha ^2} + {\beta ^2}} \over {{\alpha ^2}{\beta ^2}}} - {b \over a}\] \[ = {{{{{b^2}} \over {{a^2}}} - {{2c} \over a}} \over {{{{c^2}} \over {{a^2}}}}}\] On simplification
2
a
2
c
=
a
b
2
+
b
c
2
2{a^2}c = a{b^2} + b{c^2}
2
a
2
c
=
a
b
2
+
b
c
2
\[ \Rightarrow {{2a} \over b} = {c \over a} + {b \over c}\] \[ \Rightarrow {c \over a},{a \over b},{b \over c}\] are in
A
.
P
.
A.P.
A
.
P
.
\[\therefore\] \[{a \over c},{b \over a},\,\,\& \,\,\] are in
H
.
P
.
H.P.
H
.
P
.
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