Question

If the roots of the equation \(\frac{a}{x-a}+\frac{b}{x-b}=1\) are equal in magnitude but opposite in sign then (a) \(\mathrm{a}-\mathrm{b}=0\) (b) \(a+b=1\) (c) \(\mathrm{a}-\mathrm{b}=1\) (d) \(\mathrm{a}+\mathrm{b}=0\)

Short answer

Answer: The relationship between 'a' and 'b' that makes the roots equal in magnitude and opposite in sign is \(a+b=0\).

Step-by-step solution

Simplify the given equation

Multiplying both sides of the equation by the denominators \((x-a)(x-b)\) to eliminate the fractions, we obtain: \begin{equation} a(x-b)+b(x-a)=(x-a)(x-b) \end{equation}

Expand and Simplify

Now, let's expand and simplify the equation: \begin{equation} ax - ab + bx - ab = x^2 - bx - ax + ab \end{equation} Combining like terms, we get: \begin{equation} x^2 - (a+b)x + ab = 0 \end{equation}

Analyze the roots of the equation

Let the roots of the equation be 'c' and '-c' (equal in magnitude and opposite in sign). By Vieta's formulas, we have: \begin{equation} c + (-c) = \frac{a+b}{1} \quad \text{and} \quad -c^2=\frac{ab}{1} \end{equation} Simplifying the first equation gives us: \begin{equation} 0 = a+b \end{equation} Simplifying the second equation gives us: \begin{equation} c^2 = -ab \end{equation}

Conclusion

Based on the derived results in Step 3, the relationship between 'a' and 'b' that makes the roots equal in magnitude and opposite in sign is: \begin{equation} a + b = 0 \end{equation} This matches option (d). Thus, the correct answer is: (d) \(a+b=0\)

Key concepts

Quadratic Equations

Quadratic equations are algebraic expressions that are founded on the highest power of the unknown variable being a square. A general form of a quadratic equation is written as:\[ ax^2 + bx + c = 0 \]In this expression, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). The equation is termed "quadratic" not only because of the squared term but also because it often represents a parabola when graphed.A fascinating feature of quadratic equations is that they can have two solutions, known as roots. The nature of these roots (whether real, complex, or duplicate) depends on the discriminant \(b^2 - 4ac\). If:

• \(b^2 - 4ac > 0\): the equation has two distinct real roots,
• \(b^2 - 4ac = 0\): there is exactly one real root, called a repeated root,
• \(b^2 - 4ac < 0\): the equation has two complex conjugate roots.

Understanding quadratic equations is essential for solving a wide range of mathematical problems, and mastering this concept helps you grasp more complex algebraic structures.

Polynomial Roots

Roots of a polynomial are the solutions where the polynomial equals zero. For quadratic equations, these are the values of \(x\) satisfying the equation \(ax^2 + bx + c = 0\). These roots can be real or complex numbers. In the case of the quadratic equation, finding roots is sometimes straightforward with methods like factoring, the quadratic formula, or completing the square.A polynomial of degree \(n\) can have up to \(n\) roots, considering multiplicity and complex roots included. The sum and product of the roots carry significant importance and can be calculated without finding the exact roots using Vieta's Formulas.For our specific context, since the roots are equal in magnitude but opposite in sign, it implies:\[ c = -c \]This simplifies to \(0\), indicating symmetrical or special conditions in the equation coefficients, such as \(a + b = 0\). Recognizing these special conditions helps find solutions more efficiently without solving completely.

Vieta's Formulas

Vieta's formulas provide a relationship between the coefficients of a polynomial and the sums, as well as the products, of its roots. They are derived from expanding and comparing the expressions in polynomial equations. For a quadratic equation given by:\[ ax^2 + bx + c = 0 \]The essential formulas for the roots \(r_1\) and \(r_2\) are:

• \(r_1 + r_2 = -\frac{b}{a}\)
• \(r_1 \cdot r_2 = \frac{c}{a}\)

In the context of roots being equal in magnitude but opposite in sign, Vieta's formulas help to form equations that illustrate the conditions of \(a + b = 0\). When the sum of the roots is zero, this ensures the polynomial’s coefficients are balanced in a way to provide symmetrical roots.Learning and applying Vieta's formulas streamline the process of solving quadratic equations without diving deep into root calculation methods. It highlights the symmetry and elegance in algebraic expressions, drawing connections between the numeric and algebraic characteristics of polynomials.