Question:
If the roots of the equation \(x^2 + px + q = 0\) are in the same ratio as those of the equation \(x^2 + lx + m = 0\), then which one of the following is correct?
यदि समीकरण \(x^2 + px + q = 0\) के मूलों का अनुपात समीकरण \(x^2 + lx + m = 0\) के मूलों के अनुपात के समान है, तो निम्नलिखित में से कौन-सा सही है?
Options:
- \(m^2 p = l^2 q\)
- \(m^2 p = q^2 l\)
- \(m^2 p^2 = l^2 q\)
- \(p^2 m = l^2 q\)
- \(m^2 p = l^2 q\)
- \(m^2 p = q^2 l\)
- \(m^2 p^2 = l^2 q\)
- \(p^2 m = l^2 q\)
Correct Answer: d)
Explanation:
Let roots of \(x^2+px+q=0\) be \(\alpha, \beta\) and roots of \(x^2+lx+m=0\) be \(\gamma, \delta\).
Given \(\dfrac{\alpha}{\beta} = \dfrac{\gamma}{\delta} \implies \dfrac{(\alpha+\beta)^2}{(\alpha-\beta)^2} = \dfrac{(\gamma+\delta)^2}{(\gamma-\delta)^2}\).
\(\dfrac{p^2}{p^2-4q} = \dfrac{l^2}{l^2-4m} \implies 1 – \dfrac{4q}{p^2} = 1 – \dfrac{4m}{l^2} \implies \dfrac{q}{p^2} = \dfrac{m}{l^2} \implies p^2m = l^2q\).
माना \(x^2+px+q=0\) के मूल \(\alpha, \beta\) हैं तथा \(x^2+lx+m=0\) के मूल \(\gamma, \delta\) हैं।
दिया है \(\dfrac{\alpha}{\beta} = \dfrac{\gamma}{\delta} \implies \dfrac{(\alpha+\beta)^2}{(\alpha-\beta)^2} = \dfrac{(\gamma+\delta)^2}{(\gamma-\delta)^2}\)।
\(\dfrac{p^2}{p^2-4q} = \dfrac{l^2}{l^2-4m} \implies \dfrac{q}{p^2} = \dfrac{m}{l^2} \implies p^2m = l^2q\)।