Understanding the Equation xy – ax – by + ab = 0

We are given the equation $\text{xy – ax – by + ab = 0}$ and asked to determine what it represents geometrically.

This is a second-degree equation in variables x and y. Second-degree equations often represent conic sections like circles, ellipses, parabolas, or hyperbolas. However, they can also represent degenerate cases such as a pair of straight lines, a point, or no locus.

Factoring the Equation to Identify the Locus

Let's try to factor the given equation:

The equation is $\text{xy – ax – by + ab = 0}$.

We can group the terms to look for common factors. Let's group the first two terms and the last two terms, or the first and third, and second and fourth. Let's try grouping the first and third, and second and fourth:

$(xy - by) + (-ax + ab) = 0$

Factor out the common term 'y' from the first group and '-a' from the second group:

$y(x - b) - a(x - b) = 0$

Now, we see a common factor of $(x - b)$ in both terms. We can factor this out:

$(x - b)(y - a) = 0$

Interpreting the Factored Equation

The equation $(x - b)(y - a) = 0$ is true if and only if either the first factor is zero or the second factor is zero (or both are zero). This means:

• $x - b = 0 \implies x = b$
• OR
• $y - a = 0 \implies y = a$

The equation $\text{xy – ax – by + ab = 0}$ is therefore satisfied by any point (x, y) such that $x = b$ or $y = a$.

The equation $x = b$ represents a vertical straight line in the xy-plane, parallel to the y-axis (assuming b is a constant). The equation $y = a$ represents a horizontal straight line in the xy-plane, parallel to the x-axis (assuming a is a constant).

Thus, the original equation represents the union of these two straight lines.

Conclusion on the Geometric Representation

Since the equation $\text{xy – ax – by + ab = 0}$ can be factored into the product of two linear equations, $(x - b)(y - a) = 0$, it represents a pair of straight lines: $x = b$ and $y = a$. These two lines are perpendicular to each other.

Based on the analysis, the equation represents a pair of straight lines.