Question

If the roots of at \(^{2}+\mathrm{bt}+\mathrm{c}=0\) are reciprocals of each other, then, (a) \(\mathrm{a}=0\) (b) \(\mathrm{b}=0\) (c) \(\mathrm{b}=\mathrm{c}\) always (d) \(\mathrm{a}=\mathrm{c}\) always

Short answer

a) a = 0 b) b = 0 c) b = c always d) a = c always Answer: d) a = c always

Step-by-step solution

Finding the sum and product of the roots

In a quadratic equation \(at^{2}+bt+c=0\), let the roots be \(\alpha\) and \(\beta\). Since the roots are reciprocals of each other, we have \(\alpha = \frac{1}{\beta}\). The sum and product of the roots are given by: Sum of roots (S) = \(\alpha + \beta = -\frac{b}{a}\) Product of roots (P) = \(\alpha \times \beta = \frac{c}{a}\)

Substituting the reciprocal relationship into the sum and product formulas

We know that the roots are reciprocals of each other, so \(\alpha = \frac{1}{\beta}\). Now, we substitute this relationship into the sum and product formulas: Sum of roots: \(\alpha + \beta = \frac{1}{\beta} + \beta = -\frac{b}{a}\) Product of roots: \(\alpha \times \beta = \frac{1}{\beta} \times \beta = 1 = \frac{c}{a}\)

Analyzing the given options based on the derived equations

Now, let's analyze each option based on the derived equations: (a) \(\mathrm{a} = 0\): If a = 0, the quadratic equation becomes \(bt+c=0\), which is a linear equation. So, this option is incorrect. (b) \(\mathrm{b} = 0\): If b = 0, then the sum of roots is 0 and \(\beta=-\frac{1}{\beta}\). Solving for \(\beta\), we obtain \(\beta^2=-1\), which has complex roots. So, this option is incorrect. (c) \(\mathrm{b} = \mathrm{c}\) always: We have the product of roots as \(\frac{c}{a}=1\). However, it doesn't imply \(b = c\) always. So, this option is incorrect. (d) \(\mathrm{a} = \mathrm{c}\) always: Since the product of roots is \(\frac{c}{a}=1\), then we have \(c = a\). So, option (d) is correct.

Key concepts

Reciprocal Roots

Understanding reciprocal roots means realizing that one root is the mathematical inverse of the other. Consider a quadratic equation of the form \(at^{2} + bt + c = 0\). If the roots \(\alpha\) and \(\beta\) are reciprocals, it must be that \(\alpha = \frac{1}{\beta}\) and \(\beta = \frac{1}{\alpha}\). This special relationship results in unique implications for the standard sums and products of these roots, which we'll explore next.

When the roots are reciprocals, multiplying them gives \(\alpha \times \beta = 1\). This is because a number times its reciprocal always equals 1. This specific product condition plays a crucial role in determining the characteristics of the polynomial equation.

Sum of Roots

The sum of the roots of a quadratic equation represents a crucial property. In the standard form of the equation \(at^{2} + bt + c = 0\), the sum of the roots \(\alpha\) and \(\beta\) is given by the formula \(-\frac{b}{a}\). This formula shows how the coefficients \(a\) and \(b\) relate to the nature of the quadratic’s solutions.

For reciprocals, we have \(\alpha + \beta = \frac{1}{\alpha} + \alpha\). Simplifying this gives \(-\frac{b}{a} = \beta + \frac{1}{\beta}\). This equation doesn't simplify to any trivial solution, indicating no immediate restrictions like in some other special root conditions, but it shows how the relationship between \(b\) and \(a\) influences the sum of the reciprocals.

Product of Roots

The product of the roots in a quadratic equation is another vital element to consider. For any quadratic of the form \(at^{2} + bt + c = 0\), the product of roots \(\alpha\) and \(\beta\) is given by \(\frac{c}{a}\). This formula links the coefficients \(a\) and \(c\) directly to the roots.

In the case of reciprocal roots, \(\alpha \times \beta = 1\), we set this equal to the product formula: \(1 = \frac{c}{a}\). This simplifies to \(c = a\), establishing a necessary condition for the relationship between these coefficients when the roots are reciprocals. Thus, a reciprocal condition directly dictates a specific numerical equality between \(c\) and \(a\).

Polynomial Equation Analysis

Analyzing polynomial equations involves understanding how the coefficients of the polynomial inform the nature of its roots. Given the equation \(at^2 + bt + c = 0\), it’s important to work through the implications of specific root conditions, such as having reciprocal roots.

Through reciprocal root conditions, we derive that if \(\alpha = \frac{1}{\beta}\), the condition \(\alpha\beta = 1\) or \(\frac{c}{a} = 1\) simplifies to \(c = a\). This insight allows us to refine our analysis further when solving quadratic equations with known reciprocal roots.

This form of analysis strengthens understanding by confirming that certain equations will always lead \(a\) and \(c\) to be equal, which can significantly streamline solving similar equations or adjusting parameters to achieve desired solutions.