Question

The roots of the equation ax² + bx + c = 0 are α, β, then equation whose roots are \(\frac{1}{\alpha},\frac{1}{\beta}\) is
1. ax² - bx + c = 0
2. bx² - ax + c = 0
3. cx² + bx + a = 0
4. by² + cy + c = 0

Answer 1

Correct Answer - Option 3 : cx² + bx + a = 0

Concept:

General Form of Quadratic Equation, ax2 + bx + c = 0 

1. Sum of roots, α + β = \(\rm \frac{-b}{a}\)
2. Products of roots, αβ = \(\rm \frac{c}{a}\)
3. The quadratic equation whose roots are α and β is given by x2 - (α + β)x + αβ = 0

Calculation:

Given

⇒ ax2 + bx + c = 0

⇒ sum of root α + β = - b/a

⇒ Products of roots αβ = c/a

Given two roots are \(\frac{1}{\alpha},\frac{1}{\beta}\) 

Sum of roots = \(\frac{1}{\alpha}+\frac{1}{\beta}\) 

Products of roots = \(\frac{1}{\alpha}\frac{1}{\beta}\)

The quadratic equation is 

⇒  x2 - x (Sum of roots) + Products of roots = 0

\(⇒ x^2 - x (\frac{1}{\alpha}+\frac{1}{\beta})+\frac{1}{\alpha}.\frac{1}{\beta} =0\)

\(⇒ x^2 - x (\frac{\alpha+\beta}{\alpha\beta})+\frac{1}{\alpha\beta}=0\)

\(⇒ x^2 - x (\frac{-b}{c})+\frac{1}{\frac{c}{a}}=0\)

\(⇒ cx^2 + bx+ a=0\)

Relation between Roots and Coefficients

• If the roots of quadratic equation ax2 + bx + c, a ≠ 0, are α and ß,then α + β = \(\rm \frac{-b}{a} = - \frac{Coefficient of\: x}{Coefficient of\: x^{2}}\) and αβ = \(\rm \frac{c}{a} = \frac{Constant term}{Coefficient of\: x^{2}}\)
• The quadratic equation whose roots are α and β is given by x2 - (α + β)x + αβ = 0