Problem 74 - A Complete Resource Book in Mathematics for JEE Main

Question

If the sum of the slopes of the lines given by \(x^{2}-\) \(2 c x y-7 y^{2}=0\) is four times their product, then \(c\) has the value (A) 1 (B) \(-1\) (C) 2 (D) \(-2\)

Step-by-Step Solution

Answer

There is a misalignment as 'c' calculated does not match provided options, check setup interpretation.

Step 1: Identify the general form of the equation

The given equation is \(x^2 - 2cxy - 7y^2 = 0\). This is a quadratic in two variables, and we can treat it as a homogeneous equation of degree 2. Such an equation can represent two straight lines through the origin.

Step 2: Find slopes using auxiliary equation

For an equation of the form \(ax^2 + 2hxy + by^2 = 0\), the slopes of the lines are given by the roots of the quadratic equation \(am^2 + 2hm + b = 0\). Here, \(a = 1\), \(2h = -2c\), and \(b = -7\). So the auxiliary equation is \(m^2 - 2cm - 7 = 0\).

Step 3: Express sum and product of slopes

The sum of the slopes \((m_1 + m_2)\) is given by \(2c\), and their product \((m_1m_2)\) is \(-7\) directly from the quadratic formula.

Step 4: Set up the given condition

According to the problem, the sum of the slopes is four times their product. Hence, the equation is \(2c = 4(-7)\).

Step 5: Solve for \(c\)

Simplifying the equation gives \(2c = -28\). Dividing both sides by 2, we find \(c = -14\). This value does not match any of the provided options. However, only the logical setup as per given is validated.

Key Concepts

Homogeneous Equations

A homogeneous equation is a type of polynomial equation where each term has the same total degree. In the context of coordinate geometry, homogeneous equations frequently represent geometrical figures such as lines or conic sections like circles or ellipses. Consider the equation of the form \( ax^2 + 2hxy + by^2 = 0 \). Here, every term adds up to 2, making it a homogeneous equation of degree 2. This particular form is interesting because it often represents two lines passing through the origin.

Understanding these equations involves looking for solutions where the variables relate proportionally. With homogeneous equations of degree 2, you can use them to find relationships between slopes in the coordinate plane. By converting these equations into simpler forms, it's possible to deduce useful geometric information, such as the slopes or angles between the lines represented by the equation.

Quadratic Equations

Quadratic equations are foundational in algebra and have the form \( ax^2 + bx + c = 0 \). In coordinate geometry, such equations can define curves, but when expressed in two variables, they often describe lines through the use of an auxiliary equation. The auxiliary equation is derived from the original quadratic by substituting the slope \( m = \frac{y}{x} \) into the equation.

• The equation \( ax^2 + 2hxy + by^2 = 0 \) represents a condition in which lines can be described by the quadratic \( am^2 + 2hm + b = 0 \).
• The solutions to this equation are called the roots, or the slopes, \((m_1, m_2)\) of the lines represented.

Each solution yields a distinct slope of the line described by the equation. Common methods such as factorizations or the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) allow us to find these slopes efficiently.

For instance, in the exercise, converting the original expression into an auxiliary equation lets you find the slopes \(m_1\) and \(m_2\) that adhere to the condition of lines’ slopes they form.

Slope of a Line

The slope of a line is a fundamental concept in coordinate geometry and measures the steepness or incline of the line. It is usually denoted by the symbol \( m \) and calculated as the change in \( y \)-coordinate divided by the change in \( x \)-coordinate (\( m = \frac{\Delta y}{\Delta x} \)). In mathematical terms, if you have two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, the slope \( m \) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Understanding slopes is crucial because they can reveal much about the behavior of lines. In homogeneous quadratic equations, solving for slopes involves understanding the geometrical setup rendered by the equation. This can be traced back to obtaining the slopes of intersecting lines via the roots of the auxiliary equation derived.

• Slopes offer a quick way to visualize angles between lines, where perpendicular lines have slopes \( m_1 \) and \( m_2 \) that satisfy \( m_1 \times m_2 = -1 \).
• For the exercise, establishing conditions such as \( 2c = 4(m_1m_2) \) helps solve problems involving sets of lines with specified relationships.

By applying these principles, one can delve deeper into analytic geometry, deriving more properties from equations representing geometrical figures.