Problem 75 - A Complete Resource Book in Mathematics for JEE Main

Question

If one of the lines given by \(6 x^{2}-x y+4 c y^{2}=0\) is \(3 x+\) \(4 y=0\), then \(c\) equals (A) 1 (B) \(-1\) (C) 3 (D) \(-3\)

Step-by-Step Solution

Answer

The value of \(c\) is 1, option (A).

Step 1: Rewrite the equation of the lines

The given equation of lines is \(6x^2 - xy + 4cy^2 = 0\). This can be viewed as a quadratic in some unknown line expressions. For simplification, split it into two line equations, \((3x + 4cy)(2x - y) = 0\).

Step 2: Provide the given line equation

We are given that one of the lines is \(3x + 4y = 0\). In comparison with the factored form of the line equation, we can assume that \(3x + 4cy = 0\) corresponds to this given line equation.

Step 3: Substitute and solve for c

To find \(c\), compare \(3x + 4cy = 0\) with \(3x + 4y = 0\). The comparison shows that \(4c = 4\), hence dividing both sides by 4 gives \(c = 1\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra and come in the standard form of \( ax^2 + bx + c = 0 \). These equations can describe parabolas on a graph and have important applications in various real-world scenarios.When interacting with conic sections, quadratic equations often embody relationships of geometric shapes like ellipses, parabolas, and hyperbolas. In the context of the original exercise, the quadratic equation involves products of line equations, which can initially seem intimidating.The exercise features the equation \( 6x^2 - xy + 4cy^2 = 0 \), which you can think of as a quadratic equation with mixed variables. Here, variables \( x \) and \( y \) show how quadratic equations can extend into multiple dimensions and interact with linear equations.

Line Equations

Line equations are all about straight lines on a graph, often expressed in the slope-intercept form \( y = mx + b \) or standard form \( Ax + By = C \). These equations are simple yet powerful tools for modeling and solving geometric problems.In our specific exercise, the given line equation \( 3x + 4y = 0 \) has been used as a component in determining the unknowns in a more complex equation. It strategically represents one of the "lines," grounding the quadratic.Recognizing the equation's role bridges linear and quadratic perspectives, highlighting the intersectional nature of mathematics. You can also manipulate line equations to solve for constraints or specific points, showing their utility in diverse problem-solving scenarios.

Factorization of Polynomials

Factorization is the process of breaking down polynomials into simpler "factors" or expressions so that when multiplied, they restore the original polynomial. This approach simplifies solving higher-degree equations by replacing them with lower-degree, more manageable pieces.In the exercise at hand, factorization helps to convert the quadratic polynomial into a product of simpler line equations, \((3x + 4cy)(2x - y) = 0\). By splitting the complex polynomial, the problem becomes easier to handle.Factorization not only aids in finding roots or solutions but also enhances our understanding of the polynomial's structure, enabling deeper insights into its components and applications. By realizing these factors, you grasp the building blocks of broader mathematics.