Problem 91 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c, p, q, r\) are three non-zero complex numbers such that \(\frac{p}{a}+\frac{q}{b}+\frac{r}{c}=1+i\) and \(\frac{a}{p}+\frac{b}{q}+\frac{c}{r}=0\), then value of \(\frac{p^{2}}{a^{2}}+\frac{q^{2}}{b^{2}}+\frac{r^{2}}{c^{2}}\) is (A) 0 (B) \(-1\) (C) \(2 i\) (D) \(-2 i\)

Step-by-Step Solution

Answer

The value is \(2i\), option C.

Step 1: Analyze Given Equations

The given equations are \( \frac{p}{a} + \frac{q}{b} + \frac{r}{c} = 1+i \) and \( \frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 0 \). These represent relationships between complex numbers \(a, b, c, p, q, r\). To use them, consider multiplying the first equation by \(abc\) to eliminate the fractions.

Step 2: Eliminate Fractions in First Equation

Multiplying the first equation by \(abc\), we have \( bc\cdot p + ac\cdot q + ab\cdot r = (1+i) \cdot abc \).

Step 3: Simplify Second Equation

The second equation \(\frac{a}{p} + \frac{b}{q} + \frac{c}{r} = 0\) can become \( \frac{(abp)(bq)(cr)}{pqr} = 0\) when simplified. However, the fraction structure is the same as having \( pqc + qra + rab = 0 \). Following this, simplify it using a common denominator.

Step 4: Use Equations to Find Relationships

Compare and combine the two equations to find useful identities. Notice that the second equation means that either \(a=b=c=0\) or that the numerators cancel each other out in some way, simplifying the problem.

Step 5: Simplify the Desired Expression

We are asked to find \( \frac{p^2}{a^2} + \frac{q^2}{b^2} + \frac{r^2}{c^2} \). Notice the symmetry in the problem, leading us to suppose all terms such as \( \frac{p}{a}, \frac{q}{b}, \frac{r}{c} \) could be reciprocals. Let's test this assumption.

Step 6: Derive Result Based on Reciprocity Assumption

Assume \( \frac{p}{a} = x\), \( \frac{q}{b} = y\), \( \frac{r}{c} = z\), giving us \( x + y + z = 1+i \) and \( xy + yz + zx = 0 \). Consider \( xy + yz + zx = 0 \) implies if \( x, y, z \) are roots of a quadratic, then a possible expression for them involves such combinations.

Step 7: Find Specific Values

The goal is \( x^2 + y^2 + z^2 \). We know \((x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy+yz+zx) \). So substituting gives \( x^2 + y^2 + z^2 = (1+i)^2 + 0 = (1+i)^2 \).

Step 8: Compute Final Value

Calculate \((1+i)^2 = 1 + 2i - 1 = 2i\). So, the value of \(\frac{p^2}{a^2} + \frac{q^2}{b^2} + \frac{r^2}{c^2} = 2i\).

Key Concepts

Algebraic Identities

Algebraic identities are fundamental tools in mathematics that allow us to simplify and solve equations. Two commonly used identities are the sum and product of roots of polynomial equations. These identities help us express sums and products in simpler forms, often making it easier to deal with complex numbers. For example, consider if we have variables related through an equation. Algebraic identities can enable us to eliminate those variables by expressing the sums or products in other terms.

In our exercise, we start with two given equations involving fractions and complex numbers, hinting at the use of identities. By multiplying equations with suitable terms, we remove fractions and reorganize expressions. Such strategic manipulation joins terms into identities, simplifying them.

• Sum of terms: like \(x+y+z\ = 1+i\).
• Relationship among products: like \(xy+yz+zx\ = 0\).

Using equations alongside identities reduces complexity and highlights relationships between variables. Hence, algebraic identities provide us with a powerful shorthand for working through complex problem structures.

Quadratic Equations

Quadratic equations form the bedrock for solving algebraic problems involving second-degree polynomials. They appear frequently in mathematics, offering solutions through roots which can be real or complex. A standard quadratic equation is in the form \(ax^2 + bx + c = 0\).

Solving these equations often involves identifying the roots using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The nature of the roots depends on the discriminant \(b^2 - 4ac\). In our context, understanding how quadratic equations factor into identifying interactions between given complex numbers is valuable.

• Simplification: Turning complex relations into quadratic forms.
• Recognizing patterns: The sum and product of roots, observed as algebraic identities.

By assuming roots are related, as we notice with the values set by identities, we can extend them into quadratic expressions. Working through these interactions, quadratic equations offer pathways to find solutions and understand the structure within the problem.

Mathematical Symmetry

Symmetry in mathematics reflects the balance or uniformity within structures, particularly when elements exhibit consistency in form after transformations. In the exercise, symmetry isn't about shapes but about the interplay of variables leading to identical expressions.

Mathematical symmetry is reflected when equations transform reciprocally, yet result in coherent expressions. For example, identifying that \(\frac{p}{a}, \frac{q}{b}, \frac{r}{c}\) can be viewed in reciprocal or symmetric contributions.

• Consistent patterns: Across varying expressions providing the same identity or result.
• Symmetrical relationships: Pervading all complex number interactions and expressions.

Approaching the exercise through symmetry helps us uncover relationships without excessive calculations, focusing instead on consistent patterns presented by symmetrical balances. Therefore, understanding symmetry protects against oversight in solving algebraically intensive problems, ensuring relatable and solvable results.