This problem involves finding the value of a constant 'c' in a quadratic equation representing a pair of straight lines. We are given a condition relating the sum and product of the slopes of these lines.

Slopes Sum and Product Analysis

The question asks us to find the value of 'c' given the equation $x^2 - 2cxy - 7y^2 = 0$. This equation represents a pair of straight lines passing through the origin. We are told that the sum of the slopes of these lines is four times their product.

Pair of Lines Equation Concepts

A general equation representing a pair of straight lines passing through the origin is given by:

$$ Ax^2 + 2Hxy + By^2 = 0 $$

If $m_1$ and $m_2$ are the slopes of these two lines, then the following relationships hold:

• Sum of slopes: $m_1 + m_2 = -\frac{2H}{B}
• Product of slopes: $m_1 \times m_2 = \frac{A}{B}

Applying Slopes Formulas

First, let's compare the given equation $x^2 - 2cxy - 7y^2 = 0$ with the general form $Ax^2 + 2Hxy + By^2 = 0$ to identify the coefficients:

• $A = 1$
• $2H = -2c \implies H = -c$
• $B = -7$

Now, we can calculate the sum and product of the slopes using the identified coefficients:

• Sum of slopes ($m_1 + m_2$):

  $$ m_1 + m_2 = -\frac{2H}{B} = -\frac{2(-c)}{-7} = \frac{2c}{-7} = -\frac{2c}{7} $$

• Product of slopes ($m_1 \times m_2$):

  $$ m_1 \times m_2 = \frac{A}{B} = \frac{1}{-7} = -\frac{1}{7} $$

Solving for the Value of c

The problem states that the sum of the slopes is four times their product. We can write this condition as:

$$ m_1 + m_2 = 4 \times (m_1 \times m_2) $$

Substitute the calculated values of the sum and product of slopes into this equation:

$$ -\frac{2c}{7} = 4 \times \left(-\frac{1}{7}\right) $$

Simplify the equation:

$$ -\frac{2c}{7} = -\frac{4}{7} $$

To solve for 'c', we can multiply both sides of the equation by 7:

$$ -2c = -4 $$

Now, divide both sides by -2:

$$ c = \frac{-4}{-2} $$

$$ c = 2 $$

Therefore, the value of c is 2.