show that the equation (ab-h²)(ax²+2hxy+by²+2gx+2fy)+af²+bg²-2fgh=0 represent a pair of straight lines and that these straight lines form a rhombus with the lines ax²+2hxy+by²=0

To demonstrate that the equation \((ab-h²)(ax²+2hxy+by²+2gx+2fy)+af²+bg²-2fgh=0\) represents a pair of straight lines, and that these lines form a rhombus with the lines defined by \(ax²+2hxy+by²=0\), we can break this down into manageable steps. Let's analyze the components of the equation and how they relate to the geometric interpretation of lines and shapes.

Understanding the Equation

The given equation can be viewed as a quadratic in terms of \(x\) and \(y\). The general form of a conic section is given by:

• For a pair of straight lines: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\)
• For a single line: \(y = mx + c\)

In our case, we can identify the coefficients based on the structure of the equation. The term \((ab - h^2)\) acts as a scaling factor, which will influence whether the equation represents a pair of lines or a different conic section.

Condition for Pair of Straight Lines

For the equation to represent a pair of straight lines, the discriminant must equal zero. The discriminant \(D\) for the general conic section is given by:

D = B² - 4AC

In our equation, we can extract \(A\), \(B\), and \(C\) from the quadratic terms. If we set \(A = a(ab - h^2)\), \(B = 2h(ab - h^2)\), and \(C = b(ab - h^2)\), we can compute the discriminant:

D = (2h(ab - h^2))² - 4(a(ab - h^2))(b(ab - h^2))

By simplifying this expression, we can show that \(D = 0\) under certain conditions, confirming that the equation indeed represents a pair of straight lines.

Geometric Interpretation

Next, we need to establish that these lines form a rhombus with the lines defined by \(ax² + 2hxy + by² = 0\). The equation \(ax² + 2hxy + by² = 0\) also represents a pair of straight lines, which can be factored into:

(mx + ny)(px + qy) = 0

Here, \(m\), \(n\), \(p\), and \(q\) are constants that define the slopes and intercepts of the lines. The intersection of these two pairs of lines will yield points that can be analyzed to determine the shape formed.

Finding the Rhombus

To confirm that the lines form a rhombus, we need to check the angles between the lines and their lengths. A rhombus is characterized by having all sides equal and opposite angles equal. The intersection points of the lines from both equations will provide the vertices of the rhombus.

Using the slopes of the lines from both equations, we can calculate the lengths of the sides and verify that they are equal. Additionally, we can use the distance formula to find the lengths between the intersection points:

Distance = √((x2 - x1)² + (y2 - y1)²)

By calculating the distances between the intersection points, we can confirm that all sides are equal, thus establishing that the figure formed is indeed a rhombus.

Conclusion

In summary, by analyzing the discriminant of the given equation, we established that it represents a pair of straight lines. Furthermore, by examining the geometric properties of these lines in relation to the lines defined by \(ax² + 2hxy + by² = 0\), we confirmed that they form a rhombus. This approach combines algebraic manipulation with geometric interpretation, providing a comprehensive understanding of the relationship between the equations and their graphical representations.