Distance between point and line

  • MHB
  • Thread starter Yankel
  • Start date
  • Tags
    Line Point
In summary, the conversation discusses finding the equation of a line passing through the origin and between two given points, where the sum of distances from the line to the points is known. The formula used is the distance formula, and after setting up the equation, it is simplified to one variable by using the fact that the point (0,0) lies on the line.
  • #1
Yankel
395
0
Hello all,

I need your assistance with the following:

Find the equation of the line that goes through the origin (0,0) if it is known that the sum of distances of the line from A(2,1) and B(1,4) is equal to \[\sqrt{8}\], and it is also known that the line is between the points.

The formula I am supposed to use is:

\[d=\frac{\left | Ax+By+C \right |}{\sqrt{A^{2}+B^{2}}}\]

What I did, is I set the equation to be:

\[\frac{-2A-B-C}{\sqrt{A^{2}+B^{2}}}+\frac{A+4B+C}{\sqrt{A^{2}+B^{2}}}=\sqrt{8}\]

The absolute value was omitted since the line is between the points, i.e. one point is below and one above.

I am stuck with 1 equation and 2 variables. How do I proceed ? Since the point (0,0) is on the line, I know that C is 0 in the equation Ax+By+C=0, but this parameter is not important anyway (C-C=0).

Thank you !
 
Mathematics news on Phys.org
  • #2
The family of lines passing through the origin may be written:

\(\displaystyle Ax+y=0\)

where the slope is $-A$. Now you will have only 1 variable in your equation. ;)
 

FAQ: Distance between point and line

What is the distance between a point and a line?

The distance between a point and a line is the shortest distance from the point to any point on the line.

How do you calculate the distance between a point and a line?

The distance between a point and a line can be calculated using the formula d = |ax + by + c| / sqrt(a^2 + b^2), where (x, y) is the coordinates of the point and ax + by + c = 0 is the equation of the line.

What is the relationship between the distance between a point and a line and the slope of the line?

The distance between a point and a line is inversely proportional to the slope of the line. This means that as the slope of the line increases, the distance between the point and the line decreases, and vice versa.

Can the distance between a point and a line be negative?

No, the distance between a point and a line is always a positive value. If the point is on the same side of the line as the normal vector, the distance will be positive. If the point is on the opposite side, the distance will be negative.

Can the distance between a point and a line be zero?

Yes, the distance between a point and a line can be zero if the point lies on the line. This means that the point and the line share the same coordinates, and therefore, the distance between them is zero.

Similar threads

Replies
19
Views
2K
Replies
2
Views
1K
Replies
1
Views
1K
Replies
4
Views
1K
Replies
20
Views
1K
Replies
11
Views
2K
Replies
4
Views
1K
Replies
1
Views
868
Back
Top