Multiple choice

The ratio between the Gm's of the roots of the quadratic equations $ax^2 + bx + c = 0$ and $\ell x^2 + mx + n = 0$ is

  1. $\pm \sqrt{\dfrac{b \ell}{an}}$
  2. $\pm \sqrt{\dfrac{c \ell}{an}}$
  3. $\pm \sqrt{\dfrac{an}{c \ell}}$
  4. $\pm \sqrt{\dfrac{cn}{a \ell}}$
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B Correct answer
Explanation

The GM of the roots of ax^2 + bx + c = 0 is sqrt(c/a). The GM of the roots of lx^2 + mx + n = 0 is sqrt(n/l). The ratio is sqrt(c/a) / sqrt(n/l) = sqrt(cl/an).

AI explanation

For the first equation ax^2 + bx + c = 0, the product of the roots is c/a. For the second equation, the product of the roots is n/l. The geometric mean of the first pair is sqrt(c/a) and the geometric mean of the second pair is sqrt(n/l). Their ratio is sqrt(c/a) divided by sqrt(n/l), which simplifies to sqrt(cl/an).

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