Problem 74 - A Complete Resource Book in Mathematics for JEE Main 2017

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

The value of \(c\) is \(-2\).

Step 1: Identify the equation of the conic

The given equation is a homogeneous quadratic equation in terms of \(x\) and \(y\): \(x^2 - 2cxy - 7y^2 = 0\). This represents a pair of straight lines passing through the origin.

Step 2: Use the standard form of conic equation to find slope

For a general second-degree equation \(ax^2 + 2hxy + by^2 = 0\) representing lines, the slopes \(m_1\) and \(m_2\) are found using the relation \(m_1 + m_2 = -\frac{2h}{b}\) and \(m_1 \times m_2 = \frac{a}{b}\).

Step 3: Apply the condition on slopes

According to the problem, the sum of slopes \(m_1 + m_2\) is four times their product \(m_1 \times m_2\). This gives the equation: \(-\frac{2(-c)}{-7} = 4 \cdot \frac{1}{-7}\).

Step 4: Simplify the equation

Simplify the condition to find: \(\frac{2c}{7} = \frac{-4}{7}\).

Step 5: Solve for \(c\)

Solve the equation \(\frac{2c}{7} = \frac{-4}{7}\) to get \(2c = -4\). Dividing both sides by 2 results in \(c = -2\).

Step 6: Conclusion

Based on the computed value, \(c\) is \(-2\), which corresponds to option (D).

Key Concepts

Pair of Straight Lines

The equation provided in the exercise, \(x^2 - 2cxy - 7y^2 = 0\), is a fascinating example of a pair of straight lines. Mathematically, this equation is known as a homogeneous quadratic equation. Such equations typically represent conic sections, and in this particular case, they indicate two lines intersecting at the origin.
The defining feature of a pair of straight lines is the presence of two distinct linear components. This occurs when a conic equation can be factored into two linear equations. For instance, the given quadratic equation signifies that there are two actual lines that meet at the point \((0,0)\).
To determine the individual slopes of these component lines, we must engage with the coefficients of the equation in the right manner. By examining the values of these coefficients, the distinct but connected equations of the lines become apparent. This aids in analyzing how the lines interact with one another, particularly their angles and intersection points.

Slope of a Line

The slope of a line is a crucial property that indicates how steep the line is. In the context of the given equation, which represents a pair of straight lines, finding the slopes is key to solving the exercise. The slope is denoted by \(m\) and calculated using the structure of the quadratic equation.
For a conic section given by the general equation \(ax^2 + 2hxy + by^2 = 0\), the slopes of the individual lines, say \(m_1\) and \(m_2\), can be found with simple formulas:

• The sum of the slopes: \(m_1 + m_2 = -\frac{2h}{b}\)
• The product of the slopes: \(m_1 \times m_2 = \frac{a}{b}\)

The problem in the exercise is solved by relating these expressions to each other, particularly using the given condition— that the sum of the slopes is four times their product. This relationship directly influences the constants in the equation and aids in computing the necessary values.

Homogeneous Quadratic Equation

A homogeneous quadratic equation, like the one presented in the exercise \(x^2 - 2cxy - 7y^2 = 0\), is identified by its degree and the nature of its terms. Each term in a homogeneous equation contributes a specific power in terms of variables \(x\) and \(y\).
In this particular form, it correlates to a conic section, but more specifically, it reveals information about linear equations when analyzed thoroughly. In other words, a homogeneous quadratic equation can sometimes break down into two linear factors, representing the pair of lines discussed earlier.
Understanding that the equation is homogeneous is significant because it demonstrates symmetry around the origin, which simplifies computations and helps us establish important relationships within the equation. For instance, the coefficients in the equation directly control the angle of intersection and other essential characteristics of the lines derived from that equation.
This is how mathematics cleverly uses symmetry and algebraic manipulation to decode complex geometrical relationships, offering solutions that might initially appear elusive.