Understanding the Condition for a Pair of Straight Lines

A general second-degree equation in two variables, like \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\), represents a pair of straight lines if and only if a specific condition related to its coefficients is met. This condition is that the determinant of the matrix formed by the coefficients must be zero. The determinant can be expressed in terms of the coefficients \(a, b, c, f, g, h\).

The Determinant Condition for a Pair of Straight Lines

The condition for the equation \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\) to represent a pair of straight lines is given by:

\(abc + 2fgh - af^2 - bg^2 - ch^2 = 0\)

Alternatively, this condition can be remembered using the determinant of the following matrix:

The determinant of this matrix must be equal to zero for the equation to represent a pair of straight lines.

Comparing the Given Equation with the General Form

The given equation is \(3x^2 + 7xy + 2y^2 + 5x + 5y + k = 0\).

Comparing this with the general equation \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\), we can identify the coefficients:

• \(a = 3\)
• \(2h = 7 \implies h = \frac{7}{2}\)
• \(b = 2\)
• \(2g = 5 \implies g = \frac{5}{2}\)
• \(2f = 5 \implies f = \frac{5}{2}\)
• \(c = k\)

Solving for k using the Condition for a Pair of Straight Lines

Now, we substitute these coefficient values into the determinant condition \(abc + 2fgh - af^2 - bg^2 - ch^2 = 0\).

The equation becomes:

\((3)(2)(k) + 2\left(\frac{5}{2}\right)\left(\frac{5}{2}\right)\left(\frac{7}{2}\right) - (3)\left(\frac{5}{2}\right)^2 - (2)\left(\frac{5}{2}\right)^2 - (k)\left(\frac{7}{2}\right)^2 = 0\)

Let's simplify each term:

• \((3)(2)(k) = 6k\)
• \(2\left(\frac{5}{2}\right)\left(\frac{5}{2}\right)\left(\frac{7}{2}\right) = 2 \times \frac{175}{8} = \frac{175}{4}\)
• \((3)\left(\frac{5}{2}\right)^2 = 3 \times \frac{25}{4} = \frac{75}{4}\)
• \((2)\left(\frac{5}{2}\right)^2 = 2 \times \frac{25}{4} = \frac{50}{4}\)
• \((k)\left(\frac{7}{2}\right)^2 = k \times \frac{49}{4} = \frac{49k}{4}\)

Substitute these simplified terms back into the condition:

\(6k + \frac{175}{4} - \frac{75}{4} - \frac{50}{4} - \frac{49k}{4} = 0\)

Combine the fractions:

\(6k + \frac{175 - 75 - 50}{4} - \frac{49k}{4} = 0\)

\(6k + \frac{50}{4} - \frac{49k}{4} = 0\)

\(6k + \frac{25}{2} - \frac{49k}{4} = 0\)

To eliminate the denominators, multiply the entire equation by 4:

\(4 \times (6k) + 4 \times \left(\frac{25}{2}\right) - 4 \times \left(\frac{49k}{4}\right) = 4 \times 0\)

\(24k + 50 - 49k = 0\)

Combine the terms with k:

\((24 - 49)k + 50 = 0\)

\(-25k + 50 = 0\)

Solve for k:

\(-25k = -50\)

\(k = \frac{-50}{-25}\)

\(k = 2\)

Thus, for the given equation to represent a pair of straight lines, the value of \(k\) must be 2.

Conclusion

By applying the condition for a general second-degree equation to represent a pair of straight lines, we found that the value of \(k\) is 2. This condition is a fundamental concept in coordinate geometry when dealing with the nature of curves represented by quadratic equations.