If the equation (1 + m2)x2 + 2mcx + c2 - a2 = 0 has equal roots then show that c2 = a2(1 +m2).
The given equation is : (1 + m2)x2 + 2mcx + c2 - a2 = 0
Roots of an equation are given by the formula:
x = 
, where, D = b2 – 4ac
For equal roots, D = 0
On comparing with ax2 + bx + c = 0, we get:
a = 1 + m2, b = 2mc, c = c2 - a2
Therefore, D = 0
⇒ (2mc)2 – 4(1 + m2)(c2 - a2) = 0
⇒ 4m2c2 – 4c2 + 4a2 – 4m2c2 + 4m2a2 = 0
⇒ - 4c2 + 4a2 + 4m2a2 = 0
⇒ 4c2 = 4a2 + 4m2a2
⇒ 4c2 = 4a2 (1 + m2)
⇒ c2 = a2 (1 + m2)
Hence, proved.
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