# Shortest distance between a line and a point?

• Helly123
In summary, the shortest distance from (0,0) to the line passing through points A(2,3) and B(3,5) is (1/5) ##\sqrt5##.
Helly123

## Homework Statement

find the shortest distance from (0,0) to the line passing A(2,3) and B(3,5)

## Homework Equations

## \frac{y-y1}{y2-y1} = \frac{x-x1}{x2-x1} ##
y-y1 = m (x-x1)
m1 * m2 = -1 (m1 perpendicular to m2)

## The Attempt at a Solution

line passing A and B points
## \frac{y-3}{5-3} ## = ## \frac{x-2}{3-2} ##
y-3 = 2(x-2)
y = 2x - 1

m1 = 2
m2 = -1/2

the line perpendicular to line 1 and passing (0,0)
y -y1 = m(x-x1)
y-0=-1/2(x-0)
y' = -1/2x'

line 1 and line 2 intersect at (x,y)
y = y', x = x'
2x - 1 = -1/2x'
3/2x = 1
x = 2/3

y' = -1/2'x = -1/2(2/3) = -1/3

x,y = 2/3 , -1/3

distance is from (0,0) to (2/3 , -1/3)
distance = ## \sqrt{(2/3-0)^2 + (-1/3-0)^2} ##
distance = ## \sqrt{4/9 + 1/9} ##
distance = ## \sqrt{5}/3 ##

what's wrong? I got wrong answer

Helly123 said:

## Homework Statement

find the shortest distance from (0,0) to the line passing A(2,3) and B(3,5)

## Homework Equations

## \frac{y-y1}{y2-y1} = \frac{x-x1}{x2-x1} ##
y-y1 = m (x-x1)
m1 * m2 = -1 (m1 perpendicular to m2)

## The Attempt at a Solution

line passing A and B points
## \frac{y-3}{5-3} ## = ## \frac{x-2}{3-2} ##
y-3 = 2(x-2)
y = 2x - 1

m1 = 2
m2 = -1/2

the line perpendicular to line 1 and passing (0,0)
y -y1 = m(x-x1)
y-0=-1/2(x-0)
y' = -1/2x'

line 1 and line 2 intersect at (x,y)
y = y', x = x'
2x - 1 = -1/2x
3/2x = 1
x = 2/3

y' = -1/2'x = -1/2(2/3) = -1/3

x,y = 2/3 , -1/3

distance is from (0,0) to (2/3 , -1/3)
distance = ## \sqrt{(2/3-0)^2 + (-1/3-0)^2} ##
distance = ## \sqrt{4/9 + 1/9} ##
distance = ## \sqrt{5}/3 ##

what's wrong? I got wrong answer
The red line.

Helly123
Helly123 said:

## Homework Statement

find the shortest distance from (0,0) to the line passing A(2,3) and B(3,5)

## Homework Equations

## \frac{y-y1}{y2-y1} = \frac{x-x1}{x2-x1} ##
y-y1 = m (x-x1)
m1 * m2 = -1 (m1 perpendicular to m2)

## The Attempt at a Solution

line passing A and B points
## \frac{y-3}{5-3} ## = ## \frac{x-2}{3-2} ##
y-3 = 2(x-2)
y = 2x - 1

m1 = 2
m2 = -1/2

the line perpendicular to line 1 and passing (0,0)
y -y1 = m(x-x1)
y-0=-1/2(x-0)
y' = -1/2x'

line 1 and line 2 intersect at (x,y)
y = y', x = x'
2x - 1 = -1/2x'
3/2x = 1
x = 2/3

y' = -1/2'x = -1/2(2/3) = -1/3

x,y = 2/3 , -1/3

distance is from (0,0) to (2/3 , -1/3)
distance = ## \sqrt{(2/3-0)^2 + (-1/3-0)^2} ##
distance = ## \sqrt{4/9 + 1/9} ##
distance = ## \sqrt{5}/3 ##

what's wrong? I got wrong answer
Simply an Algebra mistake. Highlighted in red.

If you graph the line ##\ y=2x-1\ ##, you will notice that the point (2/3, −1/3) does not lie on the line.

Last edited:
Helly123
Helly123 said:
y' = -1/2x'
.
.
.
2x - 1 = -1/2x'
3/2x = 1
@ehild already pointed out the mistake in the last line above. In addition, the three lines I picked out above are ambiguous, as -1/2x' might be interpreted as ##-\frac{1}{2x'}## by some when you probably meant ##-\frac{1}2 x'##. The same goes for 3/2x.

Helly123
thank you for all the corrections... :)

Helly123 said:
thank you for all the corrections... :)

SammyS said:
(1/5) ##\sqrt5##

SammyS

## 1. What is the formula for finding the shortest distance between a line and a point?

The formula for finding the shortest distance between a line and a point is: d = |ax + by + c| / √(a^2 + b^2), where (x,y) is the coordinates of the point and ax + by + c = 0 is the equation of the line.

## 2. Can the shortest distance between a line and a point be negative?

No, the shortest distance between a line and a point is always a positive value. This is because the absolute value function is used in the formula to ensure a positive result.

## 3. How does the slope of the line affect the shortest distance between a line and a point?

The slope of the line does not affect the shortest distance between a line and a point. The distance is only dependent on the coordinates of the point and the equation of the line.

## 4. Is there a geometric interpretation of the shortest distance between a line and a point?

Yes, the shortest distance between a line and a point is the length of the perpendicular line from the point to the line. This can be visualized as a right triangle formed by the point, the perpendicular line, and the point where the perpendicular line meets the original line.

## 5. Can the shortest distance between a line and a point be greater than the distance between the point and the origin?

Yes, it is possible for the shortest distance between a line and a point to be greater than the distance between the point and the origin. This can occur if the point is very close to the line, but not on it, resulting in a shorter distance to the origin than the distance to the line.

• Precalculus Mathematics Homework Help
Replies
17
Views
1K
• Precalculus Mathematics Homework Help
Replies
7
Views
958
• Precalculus Mathematics Homework Help
Replies
6
Views
259
• Precalculus Mathematics Homework Help
Replies
2
Views
971
• Precalculus Mathematics Homework Help
Replies
6
Views
1K
• Precalculus Mathematics Homework Help
Replies
10
Views
664
• Precalculus Mathematics Homework Help
Replies
5
Views
912
• Precalculus Mathematics Homework Help
Replies
7
Views
2K
• Precalculus Mathematics Homework Help
Replies
14
Views
511
• Precalculus Mathematics Homework Help
Replies
2
Views
579