Key Concepts and Formulas

• Homogeneous Equation of Second Degree: An equation of the form $ax^2 + 2hxy + by^2 = 0$ represents a pair of straight lines passing through the origin.
• Sum of Slopes: If $m_1$ and $m_2$ are the slopes of the lines represented by $ax^2 + 2hxy + by^2 = 0$, then $m_1 + m_2 = -\frac{2h}{b}$.
• Product of Slopes: If $m_1$ and $m_2$ are the slopes of the lines represented by $ax^2 + 2hxy + by^2 = 0$, then $m_1 m_2 = \frac{a}{b}$.

Step-by-Step Solution

Step 1: Identify the coefficients from the given equation. The given equation is $x^2 - 2cxy - 7y^2 = 0$. We need to compare this with the general form $ax^2 + 2hxy + by^2 = 0$ to find the values of $a$, $h$, and $b$.

• $a = 1$ (coefficient of $x^2$)
• $2h = -2c$, so $h = -c$ (coefficient of $xy$)
• $b = -7$ (coefficient of $y^2$)

Step 2: Calculate the sum of the slopes ($m_1 + m_2$). We use the formula $m_1 + m_2 = -\frac{2h}{b}$. Substituting the values of $h$ and $b$ from Step 1: $m_1 + m_2 = -\frac{2(-c)}{-7} = -\frac{2c}{7}$

Step 3: Calculate the product of the slopes ($m_1 m_2$). We use the formula $m_1 m_2 = \frac{a}{b}$. Substituting the values of $a$ and $b$ from Step 1: $m_1 m_2 = \frac{1}{-7} = -\frac{1}{7}$

Step 4: Apply the given condition. The problem states that the sum of the slopes is four times their product. This means: $m_1 + m_2 = 4(m_1 m_2)$ Substituting the expressions we found in Steps 2 and 3: $-\frac{2c}{7} = 4\left(-\frac{1}{7}\right)$

Step 5: Solve for $c$. We now have an equation with only $c$ as the unknown. We need to solve for $c$: $-\frac{2c}{7} = -\frac{4}{7}$ Multiply both sides by 7 to eliminate the fractions: $-2c = -4$ Divide both sides by -2 to isolate $c$: $c = \frac{-4}{-2} = 2$

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs, especially when identifying $h$ from $2h$ and when applying the formulas for the sum and product of slopes. A single sign error can lead to an incorrect answer.
• Coefficient Identification: Make sure you correctly identify the coefficients $a$, $h$, and $b$ by comparing the given equation with the general form.
• Formula Application: Double-check that you are using the correct formulas for the sum and product of slopes.

Summary

We identified the coefficients from the given equation, calculated the sum and product of the slopes using the standard formulas, and applied the given condition that the sum of the slopes is four times their product. Solving the resulting equation for $c$ gave us $c=2$.

Final Answer The final answer is \boxed{2}, which corresponds to option (C).