Find the point on the line 1x+7y-4=0 which is closest to the point (-4,-6)
To find the point on a line closest to a given point, you can use the concept of perpendicular distance. The point of intersection between the line and the perpendicular line through the given point will be the closest point on the line.
Step 1: Find the slope of the line
Rewrite the equation of the line in slope-intercept form (y = mx + c), where m is the slope of the line.
1x + 7y - 4 = 0
7y = -1x + 4
y = (-1/7)x + 4/7
From this equation, we can see that the slope of the line is -1/7.
Step 2: Find the slope of the perpendicular line
The slope of the line perpendicular to the given line is the negative reciprocal of its slope. So, the slope of the perpendicular line will be 7 (reciprocal of -1/7).
Step 3: Find the equation of the perpendicular line
Using the point-slope form of a line, we have:
y - y1 = m(x - x1)
where (x1, y1) are the coordinates of the given point (-4, -6) and m is the slope of the perpendicular line (7).
y + 6 = 7(x + 4)
y + 6 = 7x + 28
y = 7x + 22
Step 4: Solve for the point of intersection
To find the point of intersection, we need to solve the system of equations formed by the given line and the perpendicular line.
Solving the system of equations:
y = (-1/7)x + 4/7
y = 7x + 22
(-1/7)x + 4/7 = 7x + 22
Multiply both sides by 7 to get rid of the denominators:
-1x + 4 = 49x + 154
50x = -150
x = -3
Substituting the value of x into one of the equations, we can find the value of y:
y = (-1/7)(-3) + 4/7
y = 3/7 + 4/7
y = 7/7
y = 1
Therefore, the point on the line 1x + 7y - 4 = 0 which is closest to the point (-4, -6) is (-3, 1).
distance will be along the perpendicular. The line perpendicular will have a slope= 7 (negative reciprocal of slope of the curve)
y=mx+b=7x+b but the point -4,-6) is on the line, so
-6=-28+b, or b=22 check that
finally, the curve and the line intersect, so
y=-x/7+4/7
y=7x+22
solve for x, y.
Now the distance formula
distance=sqrt[(x+4)^2+(y+6)^2 using x,y above.
isn't x,y above just -4,-6 ?
what do i do with the distance formula?