Multiple choice

The roots of the equation are $lx^2 + nx+n=0$ are in the ratio $p:q$. Then

  1. $\displaystyle\sqrt{ \frac {p} {q}} +\sqrt{\frac {q} {p}} +\sqrt{\frac {n} {l}} =0$
  2. $\displaystyle\sqrt{ \frac {p} {q}} -\sqrt{\frac {q} {p}} +\sqrt{\frac {n} {l}} =0$
  3. $\displaystyle \sqrt{\frac {p} {q}} -\sqrt{\frac {q} {p}} -\sqrt{\frac {n} {l}} =0$
  4. $\displaystyle \sqrt{\frac {p} {q}} +\sqrt{\frac {q} {p}} +\sqrt{\frac {n} {l}} =1$
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A Correct answer
AI explanation

Assume the roots are p*m and q*m, then their sum is m(p+q) = -n/l and their product is m^2(p*q) = n/l. Multiplying the sum equation by m gives m^2(p+q) = -mn/l. Substituting m^2(p*q) for n/l gives p+q = -m(p+q) / m(p*q) = -1/m, so m = -1/(p+q). By writing the sum of roots as -n/l = m(p+q) and replacing m, we eventually derive that sqrt(p/q) + sqrt(q/p) + sqrt(n/l) = 0.

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