Nature of Pair of Straight Lines

In the last section, you saw how \( ax^2 + 2hxy + by^2 = 0 \) can be factorized to:

\[ \left(ax + (h + \sqrt{h^2 - ab})y\right)\left(ax + (h - \sqrt{h^2 - ab})y\right) = 0 \]

where each factor represents a straight line.

Observe that the lines are distinct only when \( h^2 - ab > 0 \). This means the discriminant is positive, leading to two distinct linear factors, which correspond to two distinct straight lines passing through the origin.

When \( h^2 - ab = 0 \), the equation becomes:

\[ \left(ax + hy\right)\left(ax + hy\right) = 0 \]

In this case, both factors are the same, implying that the quadratic equation represents two coincident straight lines, i.e., a single straight line counted twice.

When \( h^2 - ab < 0 \), the square root term becomes imaginary, and the equation cannot be factored into real linear factors. This implies that no real straight lines exist, and the quadratic equation represents a degenerate conic section, such as an ellipse, parabola, or hyperbola.

In summary,

\( h^2 - ab > 0 \): The quadratic equation represents two distinct straight lines.
\( h^2 - ab = 0 \): The quadratic equation represents two coincident straight lines (the same line repeated).
\( h^2 - ab < 0 \): The quadratic equation does not represent any real straight lines.

Extracting Slopes from the Homogeneous Equation of a Pair of Straight Lines

Suppose \( ax^2 + 2hxy + by^2 = 0 \) (equation 1) is the homogeneous equation representing two straight lines \( y = m_1x \) and \( y = m_2x \). Using these forms, we can write the joint equation as:

\[ (m_1x - y)(m_2x - y) = 0 \]

Expanding this, we get:

\[ m_1m_2x^2 - (m_1 + m_2)xy + y^2 = 0 \]

(equation 2)

Now, both equation 1 and equation 2 represent the same pair of straight lines, implying that their coefficients are proportional. When two polynomials have the same roots, their coefficients must be proportional.

Thus, we compare the coefficients of \(x^2\), \(xy\), and \(y^2\) from equations 1 and 2:

\[ \frac{m_1m_2}{a} = \frac{-(m_1 + m_2)}{2h} = \frac{1}{b} \]

From these proportional relationships, we derive:

\[ m_1 + m_2 = -\frac{2h}{b} \]

\[ m_1m_2 = \frac{a}{b} \]

In most of the problems, we will see that we do not need the exact values of \( m_1 \) and \( m_2 \). Instead, their sum and product are often sufficient. You should remember these relationships:

\[ m_1 + m_2 = -\frac{2h}{b} \]

\[ m_1m_2 = \frac{a}{b} \]

These relationships allow us to analyze and solve many problems involving the pair of straight lines represented by the homogeneous equation \( ax^2 + 2hxy + by^2 = 0 \) without needing to explicitly find \( m_1 \) and \( m_2 \).