Coordinate geometry is a branch of mathematics that combines algebra and geometry to study the properties of geometric figures using numerical coordinates. In this context, understanding the concept of the slope of a line is crucial. The slope of a line, denoted by ‘m,’ is a measure of how steep or slanted a line is on a coordinate plane.
Slope Formula: The slope of a line passing through two points, (x₁, y₁) and (x₂, y₂), can be calculated using the following formula:
m= y2−y1/x2−x1
Here, ‘m’ represents the slope, and (x₁, y₁) and (x₂, y₂) are the coordinates of two distinct points on the line.
Interpreting Slope:
The slope can tell us a lot about a line:
1. Direction: A positive slope (m > 0) indicates that the line slants upwards from left to right. Conversely, a negative slope (m < 0) implies a downward slant from left to right. A zero slope (m = 0) means that the line is horizontal.
2. Steepness: The slope’s absolute value indicates the line’s steepness. Larger absolute values represent steeper lines.
3. Parallel Lines: Two lines are parallel if and only if they have the same slope.
Finding the Value of K:
Often, you may encounter equations of lines with an unknown constant ‘K.’ These equations are usually written in slope-intercept form:
y=mx+K
Here, ‘m’ represents the slope, and ‘K’ is the y-intercept, where the line crosses the y-axis.
To find the value of ‘K,’ you’ll need either the coordinates of a point on the line or more information about the line itself. If you have a point (x, y) on the line, you can substitute these values into the equation and solve for ‘K’:
y=mx+K
y=mx+K
K=y−mx
Alternatively, if you know the slope ‘m’ and the coordinates of a point (x, y) on the line, you can use the slope-intercept form to find ‘K.’
Solution:
The angle bisector is formed by the acute angle at the origin by the straight line.
Y=x and y=3x have equation y=kx then K?
Y=mx+c
Y=x
The slob of the line is 1
Formula:
Tan(A+B)=
Tan(A-B)=
Slob of Ab =m1=1
Slob of CD=m2=3
Slob of MN=K
Another important point is that the angle DOB is α; find value of alfa in the next part of the solution:
Tanα=
Now apply formula A-B
So we can write
Y=3x
Tan(45+α)=3
Tanα=
Tanα=
Tanα=
Tanα==1
We use this equation later
Slob of the MN
K= Tan= ( 45+α/2)
Click Here to check the next solution:
Understanding coordinate geometry and the concept of slope is essential for various applications in mathematics and science. Whether graphing linear equations or analyzing real-world problems, a solid grasp of these principles will help you navigate the world of coordinates and lines effectively.
