If the roots of the equation lx 2 + mx + m = 0 are in the ratio p ∶ q, then \(\sqrt {\frac{p}{q}} + \sqrt {\frac{q}{p}} {\rm{\;}} + \sqrt {\frac{m}{l}} \) is equal to
0
Solving Quadratic Roots Ratio Problem
The given equation is a quadratic equation: \(lx^2 + mx + m = 0\).
Let the roots of this equation be \(\alpha\) and \(\beta\). For a standard quadratic equation \(ax^2 + bx + c = 0\), the sum and product of roots are given by:
- Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
- Product of roots: \(\alpha \beta = \frac{c}{a}\)
In our given equation, \(a=l\), \(b=m\), and \(c=m\). Therefore, the sum and product of roots are:
- Sum of roots: \(\alpha + \beta = -\frac{m}{l}\)
- Product of roots: \(\alpha \beta = \frac{m}{l}\)
We are given that the roots are in the ratio \(p : q\). This means:
\[ \frac{\alpha}{\beta} = \frac{p}{q} \]We need to find the value of the expression \(\sqrt {\frac{p}{q}} + \sqrt {\frac{q}{p}} {\rm{\;}} + \sqrt {\frac{m}{l}}\).
Substitute the ratio of roots into the first two terms of the expression:
\[ \sqrt {\frac{p}{q}} + \sqrt {\frac{q}{p}} = \sqrt {\frac{\alpha}{\beta}} + \sqrt {\frac{\beta}{\alpha}} \]Combine these two terms by finding a common denominator:
\[ \sqrt {\frac{\alpha}{\beta}} + \sqrt {\frac{\beta}{\alpha}} = \frac{\sqrt{\alpha}}{\sqrt{\beta}} + \frac{\sqrt{\beta}}{\sqrt{\alpha}} = \frac{(\sqrt{\alpha})^2 + (\sqrt{\beta})^2}{\sqrt{\alpha} \sqrt{\beta}} = \frac{\alpha + \beta}{\sqrt{\alpha \beta}} \]Now substitute the expressions for the sum of roots (\(\alpha + \beta\)) and the product of roots (\(\alpha \beta\)) that we found from the quadratic equation:
\[ \frac{\alpha + \beta}{\sqrt{\alpha \beta}} = \frac{-\frac{m}{l}}{\sqrt{\frac{m}{l}}} \]Simplify this expression:
\[ \frac{-\frac{m}{l}}{\sqrt{\frac{m}{l}}} = \frac{-\frac{m}{l}}{\left(\frac{m}{l}\right)^{1/2}} \]Using the property of exponents \(\frac{a^x}{a^y} = a^{x-y}\):
\[ \frac{-\left(\frac{m}{l}\right)^1}{\left(\frac{m}{l}\right)^{1/2}} = - \left(\frac{m}{l}\right)^{1 - \frac{1}{2}} = - \left(\frac{m}{l}\right)^{1/2} = -\sqrt{\frac{m}{l}} \]So, we found that \(\sqrt {\frac{p}{q}} + \sqrt {\frac{q}{p}} = -\sqrt{\frac{m}{l}}\).
Now substitute this result back into the original expression we needed to evaluate:
\[ \left(\sqrt {\frac{p}{q}} + \sqrt {\frac{q}{p}}\right) + \sqrt {\frac{m}{l}} = \left(-\sqrt{\frac{m}{l}}\right) + \sqrt{\frac{m}{l}} \] \[ = 0 \]Thus, the value of the expression \(\sqrt {\frac{p}{q}} + \sqrt {\frac{q}{p}} {\rm{\;}} + \sqrt {\frac{m}{l}}\) is 0.
Revision Table: Quadratic Roots Properties
| Property | Formula for \(ax^2 + bx + c = 0\) | Formula for \(lx^2 + mx + m = 0\) |
|---|---|---|
| Sum of Roots (\(\alpha + \beta\)) | \(-\frac{b}{a}\) | \(-\frac{m}{l}\) |
| Product of Roots (\(\alpha \beta\)) | \(\frac{c}{a}\) | \(\frac{m}{l}\) |
| Ratio of Roots (\(\alpha : \beta\)) | \(\frac{\alpha}{\beta}\) | \(\frac{p}{q}\) (given) |
Additional Information: Roots in Ratio
When the roots of a quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(p:q\), we can write the roots as \(k p\) and \(k q\) for some constant \(k\). Using the sum and product of roots:
- Sum: \(kp + kq = k(p+q) = -\frac{b}{a}\)
- Product: \((kp)(kq) = k^2 pq = \frac{c}{a}\)
Dividing the square of the sum by the product gives:
\[ \frac{(k(p+q))^2}{k^2 pq} = \frac{(-b/a)^2}{c/a} \] \[ \frac{k^2 (p+q)^2}{k^2 pq} = \frac{b^2/a^2}{c/a} \] \[ \frac{(p+q)^2}{pq} = \frac{b^2}{a^2} \cdot \frac{a}{c} = \frac{b^2}{ac} \] \[ \frac{p^2 + 2pq + q^2}{pq} = \frac{b^2}{ac} \] \[ \frac{p}{q} + 2 + \frac{q}{p} = \frac{b^2}{ac} \]This identity is useful for problems involving roots in a specific ratio. In this particular problem, we used a direct substitution approach relating the ratio to the roots and then to the sum and product of roots, which also effectively solves the problem.
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