if l1 and l2 are some real linear function of x and y over real numbers and let a and b be some non zero real constants. if ABCD is a parallelogram with equations l1=0,l2+b=0,l1+a =0 and l2=0 as the equations of the side AB,BC,CD and DA respectively then what is the equation of diagonal BD?

To find the equation of diagonal BD in the parallelogram ABCD defined by the given linear equations, we first need to understand the relationships between these lines and how they define the vertices of the parallelogram. The equations provided are:

• Line AB: l1 = 0
• Line BC: l2 + b = 0
• Line CD: l1 + a = 0
• Line DA: l2 = 0

From these equations, we can derive the coordinates of the vertices of the parallelogram. Let's break down the steps to find the equation of diagonal BD.

Identifying the Lines

We have two linear functions, l1 and l2, which can be expressed in the general form:

• l1 = mx + c1
• l2 = nx + c2

Here, m and n are the slopes of the lines, while c1 and c2 are the y-intercepts. The equations of the sides of the parallelogram can be rewritten as:

• AB: y = 0 (since l1 = 0)
• BC: y = -b (from l2 + b = 0)
• CD: y = -a (from l1 + a = 0)
• DA: y = 0 (since l2 = 0)

Finding the Vertices

Now, let's determine the coordinates of the vertices:

• Point A (intersection of AB and DA): (x1, 0)
• Point B (intersection of AB and BC): (x1, -b)
• Point C (intersection of BC and CD): (x2, -b)
• Point D (intersection of DA and CD): (x2, 0)

Here, x1 and x2 are the x-coordinates where the lines intersect. Since AB and DA are horizontal lines at y = 0, the y-coordinates for points A and D are both 0. Points B and C are at y = -b and y = -a, respectively.

Equation of Diagonal BD

To find the equation of diagonal BD, we need the coordinates of points B and D:

• Point B: (x1, -b)
• Point D: (x2, 0)

The slope (m) of line BD can be calculated using the formula:

m = (y2 - y1) / (x2 - x1) = (0 - (-b)) / (x2 - x1) = b / (x2 - x1)

Using the point-slope form of the equation of a line, we can express the equation of diagonal BD as:

y - y1 = m(x - x1)

Substituting the coordinates of point B and the slope:

y - (-b) = (b / (x2 - x1))(x - x1)

Rearranging this gives us:

y + b = (b / (x2 - x1))(x - x1)

Thus, the equation of diagonal BD can be expressed in standard form. This equation represents the line that connects points B and D, completing our analysis of the parallelogram defined by the given linear equations.