If two roots of the equation (1 + m2)x2 + 2mcx + (c2 – a2) = 0, are equal, then (A) a2 = c2 (1 + m2) (B) m2 = a2 (1 + c2) (C) c2 = a2 (1 + m2) (D) am = cm
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Answer:
Given equation is (1+m
2
)x
2
+2mcx+c
2
−a
2
=0
We need to prove c
2
=a
2
(1+m
2
)
The roots are real and equal Δ=0.
Therefore, b
2
−4ac=0
⇒(2mc)
2
−4(1+m
2
)(c
2
−a
2
)=0
⇒4m
2
c
2
−4(c
2
−a
2
+m
2
c
2
−m
2
a
2
)=0
⇒m
2
c
2
−(c
2
−a
2
+m
2
c
2
−m
2
a
2
)=0
⇒m
2
c
2
−c
2
+a
2
−m
2
c
2
+m
2
a
2
=0
⇒c
2
=a
2
+a
2
m
2
⇒c
2
=a
2
(1+m
2
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