Equation of the Family of Lines

At the most, two distinct lines cross at one point. We need the general form of the two equations, which is written as to find the intersection of two lines. If the lines are not parallel, they will only intersect. A pair of scissors, a folding chair, a road cross, a signboard, and other everyday objects are examples of intersecting lines.

Point of intersection

The point of intersection is where a point crosses two lines or curves in mathematics. In Euclidean geometry, the intersection of lines might be an empty set, a point, or a line. For two lines to meet, they must both be in the same plane and not skew lines. The intersection formula determines where these lines intersect.

The intersecting lines are formed when two lines share exactly one common point. A common point connects the intersecting lines. The point of intersection is the common point that exists on all intersecting lines. There will be an intersection point between the two non-parallel straight lines that are co-planar. The point of intersection is where lines A and B cross at point O.

Family of lines

Consider the following scenario. We are given two lines, L1 and L2, and we must determine where they intersect. Solving two simultaneous linear equations is required to evaluate the point of intersection.

Let’s write the equations for the two lines in general form:

A1X + B1Y + C1 = 0

A2X + B2Y + C2 = 0

A family of lines is a group of lines that share one or two characteristics. Straight lines can be classified into two families: those with the same slope and those with the same y-intercept.

Think about the two straight lines.

A1X + B1Y + C1 = 0 ……………….(1)

A2X + B2Y + C2 = 0 ………………..(2)

The equation of the form for every nonzero constant k

A1X + B1Y + C1+ k (A2X + B2Y + C2) = 0 ………….(3)

A straight line equation is linear in both x and y.

If (x1,y1) is the point of intersection of lines 1 and 2, then both equations 1 and 2 must be satisfied:

A1X + B1Y + C1 = 0 ……………….(4)

A2X + B2Y + C2 = 0 ……………….(5)

Then we look to see if the point (x1,y1) is on 3 or not. In equation 3, we substitute x with x1 and y with y1, and we get

A1X + B1Y + C1 + k (A2X + B2Y + C2) = 0 ……………….(6)

We derive the following conclusion by combining equations 4 and 5 in equation 6:

0+k(0)=0

This demonstrates that equation 6 holds for every k and x=x1,y=y1. For any k, the point (x1,y1) lies on 6. The equation of the line across the point of intersection of lines 1 and 2 is represented by equation 6. Equation 6 indicates that there will be an endless number of lines across the place of intersection of lines 1 and 3 because k is any real number.

Characteristics of intersecting lines

• Two or more intersecting lines always meet at a single location.

• Any angle can be used to span the intersecting lines. This angle is usually larger than 0 but less than 180 degrees.

• A pair of vertical angles is formed by two intersecting lines. The vertical angles have the same vertex and are opposite angles (which is the point of intersection).

• If two straight lines are not parallel, they will intersect at a point. The point of intersection is where two straight lines meet. The intersection point can be obtained by solving both equations at the same time if two crossing straight lines have the same equations. 

• Straight lines cross at a single point on a two-dimensional graph, which is characterised by a single set of display style x-coordinates and display style y-coordinates. Because both lines travel through that place, you know the display style x-coordinates and display style y-coordinates must satisfy both equations.

Conclusion

When two lines have exactly one common point, they form intersecting lines. The intersecting lines are linked by a common point. The common point that exists on all intersecting lines is known as the point of intersection. There will be a location where the two co-planar non-parallel straight lines intersect. The intersection point is point O, where lines A and B cross.