Distance of a point from a line

Distance of a point from a line The distance of a point from a line is the distance from the point to the closest point on the line. We can find this di...

Distance of a point from a line

The distance of a point from a line is the distance from the point to the closest point on the line. We can find this distance using various methods, including the use of coordinates and the slope of the line.

Coordinate method:

Suppose a point $P(x_1, y_1)$ lies on a line with equation $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. The distance $d$ from the point $P$ to the line is given by the formula:

$d = \left|x_1 - x\right| + \left|y_1 - y\right|$

where $x$ and $y$ are the coordinates of any point on the line.

Slope method:

The slope of a line is a measure of its steepness. It is defined as the ratio of the change in y to the change in x. If the slope of a line is positive, it means that the points on the line move upwards from left to right. If the slope is negative, the points on the line move downwards from left to right.

Using the slope, we can calculate the distance from a point $P(x_1, y_1)$ to the line as follows:

$d = \frac{|y_2 - y_1|}{m}$

where $m$ is the slope of the line.

Examples:

Find the distance of point $A(2, 7)$ from the line $y = 3x - 1$ .

Using the coordinate method, we get:

Find the distance of point $B(4, 6)$ from the line $y = 2x + 1$ .

Using the slope method, we get:

$d = \frac{|6 - 2}{2} = 4$

Conclusion:

The distance of a point from a line is the distance from the point to the closest point on the line. The distance can be found using various methods, including the use of coordinates and the slope of the line