Find the point on the line 6x + y = 9 that is closest to the point (-3,1).
Solution: We need to minimize the function
d =
√((x - (-3))^2 + (y - 1)^2)
which represents the distance between the point (-3, 1) and any arbitrary point (x, y) on the line 6x + y = 9. To find the point on the line that is closest to (-3, 1), we need to minimize this distance function.
To begin, let's substitute y in terms of x from the equation of the line:
6x + y = 9
y = 9 - 6x
Now, substitute this expression for y in the distance function:
d = √((x - (-3))^2 + (9 - 6x - 1)^2)
d = √((x + 3)^2 + (8 - 6x)^2)
To minimize the distance function, we can take the derivative of d with respect to x and set it equal to zero:
d' = 0
To simplify the algebra, let's square both sides of the equation:
d^2 = (x + 3)^2 + (8 - 6x)^2
Now, let's take the derivative of d^2 with respect to x:
d^2' = 0
2(x + 3) + 2(8 - 6x)(-6) = 0
2x + 6 - 12(8 - 6x) = 0
2x + 6 - 96 + 72x = 0
74x - 90 = 0
74x = 90
x = 90/74
x = 45/37
Now that we have the x-coordinate, we can substitute it back into the equation of the line to find the y-coordinate:
y = 9 - 6(45/37)
y = 9 - 270/37
y = 333/37
Therefore, the point on the line 6x + y = 9 that is closest to the point (-3, 1) is approximately (45/37, 333/37).