Multiple choice

If roots of the equation $x^{2}(1+m^{2})+2mcx+c^{2}-a^{2}=0$ are equal, then value of $c$ is

  1. $a\ \sqrt{(1+m^{2})}$
  2. $a\ \sqrt{(1-m^{2})}$
  3. $m\ \sqrt{(1+a^{2})}$
  4. $m\ \sqrt{(1-a^{2})}$
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A Correct answer
Explanation

For a quadratic equation to have equal roots, the discriminant D = b^2 - 4ac must be 0. Here, (2mc)^2 - 4(1+m^2)(c^2 - a^2) = 0. Simplifying this leads to 4m^2c^2 - 4(c^2 - a^2 + m^2c^2 - m^2a^2) = 0, which reduces to c^2 = a^2(1+m^2). Thus, c = a*sqrt(1+m^2).

AI explanation

For equal roots, the discriminant of the quadratic equation must be zero, so we set D = b^2 - 4ac = 0. Substituting the given coefficients gives (2mc)^2 - 4(1+m^2)(c^2-a^2) = 0. Expanding and simplifying this relationship yields 4m^2c^2 - 4(c^2 - a^2 + m^2c^2 - m^2a^2) = 0, which reduces to 4a^2(1+m^2) - 4c^2 = 0. Solving for c gives the final result of a times the square root of (1+m^2).

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