Find the point on the line 7x+1y−5=0 which is closest to the point (−2,−6).
7x+1y−5=0
y=-7x+5
so the distance minimum will be on a line thru the point (-2,6) and has a slope of 1/7
y=1/7 x+b
-2=1/7 (-2)+b
b=-2+2/7
now look at the lines to find the intersection.
y=-7x+5
y=x/7 -2+2/7
set them equal
-7x+5=x/7-2+2/7
solve for x, then go back and solve for y. That (x,y) is the point on the original line which is closest to (-2,-6)
The distance from (-2,-6) to the line 7x+y-5=0 is
|7(-2)+1(-6)-5|/√(7^2+1^2) = 25/√50 = 5/√2
so, we need to find (x,y) on the line such that
(x+6)^2 + (y+2)^2 = 25/2
(x+6)^2 + (11-7x)^2 = 25/2
x = 3/2
so, y=5-7x = -11/2
So, (3/2,-11/2) is the closest point to (-2,-6)
check: the slope of the line joining those two points is
(1/2)/(7/2) = 1/7
as shown above
To find the point on the line 7x + y - 5 = 0 that is closest to the point (-2, -6), we need to follow these steps:
Step 1: Find the equation of a line perpendicular to the given line.
- Remember that perpendicular lines have slopes that are negative reciprocals of each other.
- The given line's equation is in the form ax + by + c = 0, where a = 7, b = 1, and c = -5. The slope of the given line is -a/b.
- Hence, the slope of the given line is -7/1 = -7.
- The slope of the perpendicular line will be the negative reciprocal of -7, which is 1/7.
- Let's call the equation of the perpendicular line as m1x + y + c1 = 0, where m1 = 1/7.
Step 2: Find the point of intersection between the given line and the perpendicular line.
- To find the point of intersection, we need to solve the system of equations formed by equating the given line and the perpendicular line.
- The system of equations is:
7x + y - 5 = 0 (Given line)
(1/7)x + y + c1 = 0 (Perpendicular line)
- Solve these equations simultaneously to find the values of x and y that represent the point of intersection.
Step 3: Find the distance between the point of intersection and the given point (-2, -6).
- Once you have the coordinates of the point of intersection, you can find the distance between that point and the given point (-2, -6) using the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2).
- Plug in the appropriate values to calculate the distance.
Step 4: Repeat steps 2 and 3 for different points on the line to find the minimum distance.
- You can choose different points on the given line and repeat steps 2 and 3 to find the minimum distance.
- The point that gives you the minimum distance from (-2, -6) is the point on the line closest to that point.
By following these steps, you should be able to find the point on the line 7x + y - 5 = 0 that is closest to the point (-2, -6).