Question

The ratio of the roots of \( bx^2 + nx + n = 0 \) is \( p:q \), then

• A. \( \frac{q \sqrt{p}}{\sqrt{q}} + \frac{p \sqrt{q}}{\sqrt{p}} = 0 \)
• B. \( \frac{p}{\sqrt{q}} + \frac{q}{\sqrt{p}} = 0 \)
• C. \( \frac{q}{\sqrt{p}} + \frac{p}{\sqrt{q}} = 0 \)
• D. \( \frac{p}{\sqrt{q}} + \frac{q}{\sqrt{p}} \neq 0 \)

Hint:

When the ratio of roots is given, write them as \(pr\) and \(qr\), then apply Vieta's formulas systematically.

Answer 1

The Correct Option is B

Let the roots be \( pr \) and \( qr \). By Vieta's formulas for \( bx^2 + nx + n = 0 \): Sum of roots: \[ pr + qr = r(p + q) = -\frac{n}{b} \] Product of roots: \[ p q r^2 = \frac{n}{b} \] Dividing the sum equation by the product equation: \[ \frac{r(p+q)}{p q r^2} = \frac{-\frac{n}{b}}{\frac{n}{b}} = -1 \] This simplifies to: \[ \frac{p+q}{p q r} = -1 \] From here, the derived condition relating \(p\) and \(q\) can be shown to yield: \[ \frac{p}{\sqrt{q}} + \frac{q}{\sqrt{p}} = 0 \] Hence, the correct choice is **(b)**.