How do you: Find the shortest distance, to two decimal places from the origin to the line x - 2y = - 8 without using the formula.
Finding the shortest distance, to two decimal places, from the point P ( 7,5) to the line 2x+3y = 18?
Thanks
The square of the distance from a point on that line, at location x, is
D^2 = (7-x)^2 + (5-y)^2
= (7-x)^2 + [-1 + (2x/3)]^2
Set the derivative dD^2/dx = 0
Solve for x and then use the equation 2x+3y = 18 to get y. Then compute D
Another way to do this is to find the equation for the line through the point P that is perpendicular to the line. Since the slope of 2x+3y = 18 is -2/3, the slope of the perpendicular is 3/2.
another way would be to use vectors.
Have you studied vectors?
There is no point to show you the method unless you know the topic.
To find the shortest distance from the origin to the line x - 2y = -8 without using a formula, you can follow these steps:
1. Solve the given equation for y to obtain the equation of the line in slope-intercept form: y = (1/2)x + 4
2. The shortest distance from the origin to the line will be the perpendicular distance, which intersects the line at a right angle. Perpendicular lines have slopes that are negative reciprocals. Therefore, the slope of the line perpendicular to the given line is -2.
3. Using the slope-intercept form of the equation, let's find the equation of the line perpendicular to the given line and passing through the origin (0,0):
y = -2x
4. Now, we have two equations:
Given line: y = (1/2)x + 4
Perpendicular line: y = -2x
5. Solve the system of equations to find the point of intersection. Equate the y-values:
(1/2)x + 4 = -2x
6. Solve for x:
(1/2)x + 2x = 0
(5/2)x = 0
x = 0
7. Substitute the value of x back into any of the two original equations to find y. Let's use the given line equation:
y = (1/2)(0) + 4
y = 4
8. The point of intersection is (0, 4).
9. To find the distance from the origin to the line, calculate the length of the line segment connecting the origin (0, 0) to the point of intersection (0, 4). This can be calculated as the length of the y-coordinate, which is equal to the distance.
10. Therefore, the shortest distance from the origin to the line x - 2y = -8 is 4 units.
To find the shortest distance, to two decimal places, from the point P (7, 5) to the line 2x + 3y = 18 without using a formula, you can make use of the perpendicular distance concept as well. Follow these steps:
1. Determine the slope-intercept form of the given line by solving the equation for y:
2x + 3y = 18
3y = -2x + 18
y = (-2/3)x + 6
2. The line perpendicular to the given line will have a slope that is the negative reciprocal of (-2/3), which is 3/2.
3. Using the slope-intercept form, find the equation of the line perpendicular to the given line and passing through point P (7, 5):
y - 5 = (3/2)(x - 7)
y - 5 = (3/2)x - (21/2)
y = (3/2)x - (21/2) + 5
y = (3/2)x - (21/2) + 10/2
y = (3/2)x - (11/2)
4. Now, we have two equations:
Given line: y = (-2/3)x + 6
Perpendicular line: y = (3/2)x - (11/2)
5. Solve the system of equations to find the point of intersection. Equate the y-values:
(-2/3)x + 6 = (3/2)x - (11/2)
6. Solve for x:
(-2/3)x - (3/2)x = - (11/2) - 6
[(-2)(2) - (3)(3)]x/3 = - (22 + 36)/2
(-4 - 9)x/3 = -58/2
(-13/3)x = -58/2
x = (58/2)/(13/3)
x = 174/13
x ≈ 13.38
7. Substitute the value of x back into any of the two original equations to find y. Let's use the given line equation:
y = (-2/3)(13.38) + 6
y ≈ 0.07
8. The point of intersection is approximately (13.38, 0.07).
9. To find the distance from point P (7, 5) to the line, calculate the length of the line segment connecting point P to the point of intersection. This can be calculated using the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(13.38 - 7)^2 + (0.07 - 5)^2]
≈ √[45.464 + 20.92]
≈ √66.384
≈ 8.15 (rounded to two decimal places)
Therefore, the shortest distance from point P (7, 5) to the line 2x + 3y = 18 is approximately 8.15 units.