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Question
If the equation \[\left( 1 + m^2 \right) x^2 + 2 mcx + \left( c^2 - a^2 \right) = 0\] has equal roots, prove that c2 = a2(1 + m2).
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Solution
The given equation \[\left( 1 + m^2 \right) x^2 + 2 mcx + \left( c^2 - a^2 \right) = 0\], has equal roots
Then prove that`c^2 = (1 + m^2)`.
Here,
`a = (1 + m^2), b = 2mc and, c = (c^2 - a^2)`
As we know that `D = b^2 - 4ac`
Putting the value of `a = (1 + m^2), b = 2mc and, c = (c^2 - a^2)`
`D = b^2 - 4ac`
` = {2mc}^2 - 4xx (1 +m^2) xx (c^2 - a^2)`
` = 4 (m^2 c^2) - 4(c^2 -a^2 + m^2c^2 - m^2 a^2)`
` = 4m^2c^2 - 4c^2 + 4a^2 - 4m^2 c^2 + 4m^2a^2`
` = 4a^2 + 4m^2 a^2 = 4c^2`
The given equation will have real roots, if D = 0
`4a^2 + 4m^2 a^2 - 4c^2 = 0`
`4a^2 + 4m^2a^2 = 4c^2`
`4a^2 + (1 + m^2 ) = 4c^2`
`a^2 (1 +m^2) = c^2`
Hence, `c^2 = a^2 (1 + m^2)`.
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