Problem 91 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If one of the lines of \(m y^{2}+\left(1-m^{2}\right) x y-m x^{2}=0\) is a bisector of the angle between the lines \(x y=0\), then \(\mathrm{m}\) is \(\quad\) [2007] (a) 1 (b) 2 (c) \(-1 / 2\) (d) \(-2\)

Step-by-Step Solution

Answer

The value of \(m\) is 1, which is option (a).

Step 1: Understand the Equation of Lines

The given equation is a homogeneous equation in the form of \(m y^{2} + (1-m^{2}) x y - m x^{2} = 0\). The line \(xy = 0\) forms two lines: \(x = 0\) and \(y = 0\). We need to find if one component line of the given equation bisects the angle between these two lines.

Step 2: Use the Condition for an Angle Bisector

For a line \(L : ax + by + c = 0\) to be an angle bisector of the angle between two lines, it must have equal inclination to those lines. Here, we associate the line's form to \( (y = mx) \) reflecting the slopes in the homogeneous equation characteristics.

Step 3: Formulation using Homogeneous Equation

To determine the equation of individual lines from \(my^{2} + (1 - m^2)xy - mx^2 = 0\), use the method of comparing it to \((x - my)(x + \, \text{{other line}}) = 0\) as it should match with the form \(xy = 0\). The slope then would cover the angle bisector property.

Step 4: Analyze the Angle Bisector Property

A line from the equation should equal the slopes \(+1\) or \(-1\), representing \(x = 0\) or \(y = 0\). The condition for angle bisection is \(m = \pm 1\) since the sum of slopes for the condition is zero for \( x=0, \ y=0\).

Step 5: Identify the Correct Option

Given that one option, namely (a) 1 satisfies the condition \(my^{2} + (1 - m^2)xy - mx^2 = 0\) such that one bisector \((x-y=0)\) implies \(m= \pm1\). The best fit is \(m=1\).

Key Concepts

Homogeneous Equation

In mathematics, a homogeneous equation is an equation that exhibits a consistent algebraic form across all its terms when scaled by a multiplicative factor. In the context of coordinate geometry, homogeneous equations typically describe high-order curves, but they can also characterize lines, especially when viewed through a projective perspective.
For example, the given homogeneous equation is \( my^{2} + (1-m^{2})xy - mx^{2} = 0 \). This form is quadratic and defines a conic section, often resulting in lines when factored appropriately.

• The term "homogeneous" signifies every term is of the same degree, here, degree 2.
• Solving involves finding specific conditions or key values, like the condition \( m = \pm 1 \) in this example, that reveals intersections and angle bisectors.
• Homogeneous equations can be equated to zero, representing lines that pass through the origin.

This equation portrays that when substituted by factoring, it can reveal linear constructs that solve as rotational symmetry lines crossing into axes.

Equation of Lines

The equation of a line in a standard linear form is \( ax + by + c = 0 \). From the original homogeneous equation, we can extract two linear components. This facilitates understanding their geometric representation and interaction with other lines or axes.
For instance, when discussing angle bisectors, knowing the standard slope-intercept forms, like \( y = mx \), gives clues about possible intersecting line behavior and their bisecting qualities.

• Lines derived from homogeneous equations often meet significant properties like symmetry, orthogonality, or angular bisectors.
• Understanding how lines relate geometrically gives insight into coordinate plane properties.
• In solving line-related problems, we often equate linear and quadratic components to solve for slope or intercept, making bisector identification possible.

This knowledge enables us to solve conditions like those in our exercise where a line from our equation acts like an angle bisector, fundamentally altering directionality influenced by a variable, here \(m\).

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows algebraic equations to represent geometric figures, which then can be analyzed using mathematical principles.
Through the exercise, coordinate geometry lets us explore the interaction of lines or shapes like conics and linear equations. For instance,\(xy = 0\) directly translates to line equations \(x=0\) and \(y=0\), both simple and significant in forming axes.

• Coordinate geometry facilitates solving spatial redundancies through algebra, for creating a visual interchangeable form.
• It engages with converting equations, like the homogeneous types, into a visual or geometric interpretation.
• Deep understanding enables solving complex geometric conditions, like line bisectors, simply using slopes and conditions, which in our example resolves to \( m \) equal to a specific value.

By using coordinate geometry, problems that involve angles, intersections, or transformations become approachable as relationships among equations.