Quadratic Equation Ratio of Roots Problem

 

❓ Question

If the ratio of the roots of:

$$lx^2-nx+n=0$$

is:

$$p:q$$

then find the correct relation.

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🖼 Question Image

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✍️ Short Concept

This question uses:

👉 Relations between roots and coefficients
👉 Ratio of roots
👉 Algebraic simplification.

Main trick:

$$\alpha+\beta \quad \text{and} \quad \alpha\beta$$

become equal here.

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🔷 Step 1 — Assume Roots 💯

Let roots be:

$$\alpha,\beta$$

For quadratic:

$$lx^2-nx+n=0$$

Using standard formulas:

$$\alpha+\beta=\frac{n}{l}$$
$$\alpha\beta=\frac{n}{l}$$

So:

$$\alpha+\beta=\alpha\beta$$

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🔷 Step 2 — Use Given Ratio

Given:

$$\frac{\alpha}{\beta}=\frac{p}{q}$$

Thus:

$$\sqrt{\frac{p}{q}} = \sqrt{\frac{\alpha}{\beta}} = \frac{\sqrt{\alpha}}{\sqrt{\beta}}$$

Similarly:

$$\sqrt{\frac{q}{p}} = \frac{\sqrt{\beta}}{\sqrt{\alpha}}$$

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🔷 Step 3 — Add the Terms

$$\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} = \frac{\sqrt{\alpha}}{\sqrt{\beta}} + \frac{\sqrt{\beta}}{\sqrt{\alpha}}$$

Taking LCM:

$$= \frac{\alpha+\beta}{\sqrt{\alpha\beta}}$$

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🔷 Step 4 — Substitute Values

Since:

$$\alpha+\beta=\frac{n}{l}$$

and

$$\alpha\beta=\frac{n}{l}$$

we get:

$$\frac{n/l}{\sqrt{n/l}} = \sqrt{\frac{n}{l}}$$

Thus:

$$\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} - \sqrt{\frac{n}{l}} =0$$

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✅ Final Answer

$$\boxed{ \sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} - \sqrt{\frac{n}{l}} =0 }$$

Hence,

$$\boxed{\text{Option (2)}}$$

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⭐ Golden JEE Insight

Whenever:

$$\alpha+\beta=\alpha\beta$$

try converting expressions into:

$$\frac{\alpha +\beta }{\sqrt{\alpha \beta }}$$

It simplifies many root-ratio questions instantly.