Question

If the ratio of roots of \(\rm ax^2+bx+c=0\) is r then the value of  \(\rm \frac{r}{(r+1)^2}\) is 
1. \(\rm \frac{c^2}{ b^2}\)
2. \(\rm \frac{ac}{ b^2}\)
3. \(\rm \frac{ b^2c}{a}\)
4. \(\rm \frac{ ba}{c}\)

Answer 1

Correct Answer - Option 2 : \(\rm \frac{ac}{ b^2}\)

Concept:

Consider a quadratic equation: ax² + bx + c = 0.

Let, α and β are the roots.

• Sum of roots = α + β = -b/a
• Product of the roots = αβ = c/a

 

Calculation:

Given: \(\rm ax^2+bx+c=0\)

Let, α and β are the roots of given quadratic equation.

So, \(\rm r= \frac{\alpha }{\beta }\)           (∵ ratio of roots is 'r')

Now, \(\rm \frac{r}{(r+1)^2}\) = \(\frac{\frac{\alpha }{\beta } }{(\frac{\alpha }{\beta }+1)^2}\)

\(\Rightarrow \frac{\frac{\alpha }{\beta } }{\frac{(\alpha +\beta)^2}{\beta^2 } }\)

\(\Rightarrow \frac{\frac{\alpha }{\beta } \times \beta^2}{{(\alpha +\beta)^2} }\)

\(\Rightarrow \frac{{\alpha }{\beta } }{{(\alpha +\beta)^2} }\)

\(\Rightarrow \rm \frac{\frac ca}{(-\frac ba)^2}\)                 (∵ α + β = -b/a and  αβ = c/a)

\(\Rightarrow \rm \frac{\frac ca\times a^2}{ b^2}\)

\(\Rightarrow \rm \frac{ac}{ b^2}\)

Hence, option (2) is correct.