Multiple choice

Let $k$ be a real number such that $k\neq 0$. If $\alpha$ and $\beta$ are non zero complex numbers satisfying $\alpha +\beta=-2k$ and $\alpha^2+\beta^2=4k^2-2k$, then a quadratic equation having $\dfrac{\alpha+\beta}{\alpha}$ and $\dfrac{\alpha+\beta}{\beta}$ as its roots is equal to

  1. $4x^2-4kx+k=0$
  2. $x^2-4kx+4k=0$
  3. $4kx^2-4x+k=0$
  4. $4kx^2-4kx+1=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Roots are (alpha+beta)/alpha and (alpha+beta)/beta. Sum = (alpha+beta)(1/alpha + 1/beta) = (alpha+beta)(alpha+beta)/(alpha*beta) = (-2k)^2 / (alpha*beta). From alpha^2+beta^2 = 4k^2-2k, (alpha+beta)^2 - 2alpha*beta = 4k^2-2k, so 4k^2 - 2alpha*beta = 4k^2-2k, alpha*beta = k. Sum = 4k^2/k = 4k. Product = (alpha+beta)^2 / (alpha*beta) = 4k^2/k = 4k. Equation: x^2 - (sum)x + product = x^2 - 4kx + 4k = 0.

AI explanation

Square the given sum (alpha + beta) = -2k to get (alpha)^2 + (beta)^2 + 2(alpha)(beta) = 4(k)^2. Equating this to the given expression (alpha)^2 + (beta)^2 = 4(k)^2 - 2k reveals that 2(alpha)(beta) equals 2k, so the product of alpha and beta is k. The new roots are (alpha + beta) / alpha and (alpha + beta) / beta, which simplifies to 1 + beta/alpha and 1 + alpha/beta. The sum of these new roots is 2 + (alpha^2 + beta^2) / (alpha * beta), which evaluates to 2 + (4(k)^2 - 2k) / k, giving 4k - 2. The product of the new roots is (1 + beta/alpha) multiplied by (1 + alpha/beta), expanding to 1 + (alpha^2 + beta^2)/(alpha * beta) + 1, which equals 2 + (4(k)^2 - 2k) / k to give 4k. The required quadratic equation is x^2 - (sum)x + (product) = 0, resulting in x^2 - 4kx + 4k = 0.

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