Multiple choice

In the $\displaystyle \triangle ABC$, $BC$ is produced to D and $\displaystyle \angle ACD= \frac{3\pi }{4}$and $\tan A, \tan B$ are roots of the equation $\displaystyle x^{2}-\lambda x+\mu =0.$ Then

  1. $\displaystyle \lambda ^{2}-\mu ^{2}=1+6\mu$
  2. $\displaystyle \lambda ^{2}-\mu ^{2}=1$
  3. $\displaystyle \lambda =\mu -1$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Angle C = 180 - 135 = 45 degrees. A + B = 135 degrees. tan(A+B) = (tanA + tanB) / (1 - tanA*tanB) = tan(135) = -1. Given roots are lambda and mu, so (lambda) / (1 - mu) = -1. lambda = -1 + mu, or lambda = mu - 1.

AI explanation

Since the exterior angle ACD equals the sum of the opposite interior angles, A + B = 3pi/4. Taking the tangent of both sides gives (tan A + tan B) / (1 - tan A tan B) = -1, and substituting the sum and product of the roots from the given quadratic equation (lambda and mu) results in lambda / (1 - mu) = -1. Solving this equation yields lambda = mu - 1.

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