Problem 95 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The pair of lines represented by \(3 a x^{2}+5 x y+\left(a^{2}-2\right) y^{2}=0\) are perpendicular to each other for (a) two values of \(a\) (b) \(\forall a\) (c) for one value of \(a\) (d) for no values of \(a\)

Step-by-Step Solution

Answer

Two values of \( a \).

Step 1: Identify Perpendicularity Condition

Two lines represented by a homogeneous equation of the form \( ax^2 + 2hxy + by^2 = 0 \) are perpendicular if \( a + b = 0 \). For our equation, we have \( a = 3a \), \( b = a^2 - 2 \), and \( 2h = 5 \), so the condition for perpendicularity becomes \( 3a + (a^2 - 2) = 0 \).

Step 2: Set Equation for Perpendicularity

Using the perpendicularity condition, our equation becomes: \[ 3a + a^2 - 2 = 0 \] Simplify it to get a quadratic equation in terms of \( a \).

Step 3: Formulate the Quadratic Equation

Combine terms to express the equation as a quadratic: \[ a^2 + 3a - 2 = 0 \] This is a standard quadratic form: \( ax^2 + bx + c = 0 \).

Step 4: Solve the Quadratic Equation

Use the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 3 \), and \( c = -2 \):\[ a = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} \]Calculate the discriminant and solve for \( a \).

Step 5: Calculate and Evaluate Roots

The discriminant \( \Delta = 3^2 - 4 \times 1 \times (-2) = 9 + 8 = 17 \), which is positive, indicating two real and distinct solutions. Solve:\[ a = \frac{-3 \pm \sqrt{17}}{2} \]This gives us two values for \( a \), confirming that there are two values of \( a \) for which the lines are perpendicular.

Key Concepts

Quadratic Equation

A quadratic equation is a mathematical expression of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). The term \( ax^2 \) shows that the variable \( x \) is squared, making the equation quadratic. Quadratic equations have various applications in algebra and geometry. In terms of geometry, they can represent curves such as parabolas, but in the context of this problem, the quadratic equation emerges from algebraic manipulation of conditions for perpendicular lines. To solve a quadratic equation, you can use the quadratic formula: - The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). - This formula can help find the values of \( x \) that satisfy the equation.- Solving quadratic equations is essential in physics, engineering, and various fields involving polynomial relations.

Perpendicularity Condition

When two lines are perpendicular, they intersect forming a right angle, which is a key concept in both geometry and algebra. The perpendicularity condition for lines represented by a homogeneous equation, like \( ax^2 + 2hxy + by^2 = 0 \), dictates that \( a + b = 0 \) for the lines to be perpendicular. Here's a quick breakdown of the scenario: - For the equation \( 3ax^2 + 5xy + (a^2 - 2)y^2 = 0 \), identify the coefficients associated with terms in the usual homogeneous form. - The condition for perpendicularity in this context becomes \( 3a + (a^2 - 2) = 0 \), signaling that the sum of the coefficients for \( x^2 \) and \( y^2 \) terms equals zero. - Applying this condition helps us derive a quadratic equation, which further leads us to find the specific values for \( a \) that allow the lines to be perpendicular.

Discriminant

The discriminant is a crucial part of understanding solutions to quadratic equations. Denoted as \( \Delta \), it is calculated within the quadratic formula parameter \( b^2 - 4ac \). The discriminant tells us about the nature and number of the roots of the quadratic equation.Key points about the discriminant: - A **positive** discriminant indicates two distinct real roots. - A **zero** discriminant means there is exactly one real root (a perfect square root). - A **negative** discriminant implies there are no real roots, but rather complex roots. In this exercise, the discriminant \( \Delta = 17 \) is positive, suggesting the quadratic equation \( a^2 + 3a - 2 = 0 \) has two distinct real solutions. These solutions represent the values of \( a \) that make the lines perpendicular, emphasizing the discriminant's role in determining root characteristics.