Problem 8 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If roots of equation \(\frac{1}{x+p}+\frac{1}{x+q}=\frac{1}{r}\) are equal in magnitude but opposite in sign, then show that \(p+q=2 r\) and prove that product of roots is \(-\frac{p^{2}+q^{2}}{2}\).

Step-by-Step Solution

Answer

To prove, expand the equation and use root properties to find that \(p+q=2r\) and \(-\frac{p^2+q^2}{2}\) is the product of the roots.

Step 1: Understand the Given Condition

The roots being equal in magnitude but opposite in sign imply that if one root is \( a \), the other must be \( -a \). This characteristic is crucial to solving the problem.

Step 2: Express the Equation in Standard Form

The given equation is \(\frac{1}{x+p} + \frac{1}{x+q} = \frac{1}{r}\). By finding the common denominator, rewrite it as \( \frac{(x+q) + (x+p)}{(x+p)(x+q)} = \frac{1}{r} \). Simplifying the numerator gives: \( \frac{2x + (p+q)}{(x+p)(x+q)} = \frac{1}{r} \). Multiply both sides by \((x+p)(x+q)\), aligning the equation to conventional quadratic form: \( 2x + (p + q) = \frac{(x+p)(x+q)}{r} \). This simplifies to \(2xr + (p+q)r = x^2 + (p+q)x + pq - pqr\).

Step 3: Rearrange the Equation

Rearranging the equation: \(x^2 + ((p+q) - 2r)x + (pq - pqr) = 0\). This is now a quadratic equation \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = (p+q) - 2r\) and \(c = pq - pqr\).

Step 4: Consider Conditions for Opposite Roots

For roots \(a\) and \(-a\), their sum \(a + (-a) = 0\). This implies \(b = 0\), which means \(p + q - 2r = 0\). So, \(p + q = 2r\).

Step 5: Find the Product of the Roots

The product of the roots \(a \cdot (-a) = -a^2\) is equivalent to \(c/a\) for a quadratic equation. Here, \(c = pq - pqr = -(p^2 + q^2)/2\) when substituted back: \(a^2 = -(p^2 + q^2)/2\). This matches the given condition about product of roots.

Key Concepts

Quadratic Equations

Quadratic equations are a type of algebraic equation with the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. The hallmark of a quadratic equation is the degree of the variable, which is 2, indicated by the \(x^2\) term. This specific structure leads to several properties and methods of solving these equations.

Key characteristics of quadratic equations include:

• They can have two real roots, one repeated real root, or a pair of complex conjugate roots.
• Quadratics can be solved using the quadratic formula, completing the square, or factoring when applicable.
• A parabola represents the graphical form of a quadratic equation, which opens upwards if \(a > 0\) and downwards if \(a < 0\).

In our example, rearranging and simplifying the given problem, we have categorized it into a quadratic form, allowing us to utilize these specific methods to find relationships between the coefficients.

Roots of Equations

The roots of an equation are the values of the variable that satisfy the equation, making it equal to zero. For quadratic equations, these roots are found by setting the equation \(ax^2 + bx + c = 0\) and typically solving through one of three methods: factoring, using the quadratic formula, or completing the square.

Key insights about the roots of quadratic equations include:

• The sum of the roots, given by \(x_1 + x_2\), can be found using \(-b/a\).
• The product of the roots \(x_1 \cdot x_2\) is \(c/a\).

In the example, we explored conditions where the roots are equal in magnitude but opposite in sign, which results in a sum of zero. This helped in deducing the equation \(p + q = 2r\) by setting the equation parameter \(b = 0\).

Symmetric Roots

Symmetric roots are a special condition where each root is a mirror reflection of the other. In particular, this happens when one root is the negative of the other. If one root is \(a\), the other is \(-a\).

Essential properties of symmetric roots include:

• The sum of symmetric roots is always zero, \(a + (-a) = 0\). This affects the coefficient of \(x\) in the quadratic equation, making it zero when expressed in the form \(ax^2 + bx + c = 0\).
• The setup simplifies the derivation of relationships between parameters, such as proving \(p + q = 2r\), by directly influencing the linear coefficient \(b = 0\).

This symmetry in the example leads to specific parameter relationships that guide solving such algebraic conditions efficiently.

Mathematical Proofs

Mathematical proofs provide a rigorous way to establish the truth of a statement using logical deduction. Proofs are essential in mathematics as they ensure that the conclusions drawn are universally valid.

In our example, we engage in a few steps to prove two important conditions:

• First, showing \(p + q = 2r\) involves recognizing that the sum of symmetric roots creates a zero for the linear term, \(b\), which directly leads to this condition.
• Secondly, proving the product of the roots yields \(-\frac{p^2 + q^2}{2}\) requires substitution and simplification, observing the specific structure of coefficients and roots' properties.

Mathematical proofs in this context require careful alignment of algebraic manipulation with the conditions presented, ensuring each step logically follows from the last, providing a reassuring foundation for understanding these algebraic equations.