Find the point on the line 6x + y = 9 that is closest to the point (-3,1).

Question

Solution: We need to minimize the function
d = sqrt((x − (−3))^2 + (y − 1)^2)
= sqrt((x + 3)^2 + (y − 1) ^2 )
and, since the point (x, y) lies on the line 6x + y = 9, we can eliminate y from the formula for d:
d = (x + 3)^2 + (9 − 6x − 1) ^2
= (x + 3)^2 + (8 − 6x)^2
It will be easier to minimize the square of the function:
D = d2 = (x + 3)^2 + (8 − 6x)^2
Then,
D

Answer 1

Yes, continue this way, find the derivative of D with respect to x.
Equate the derivative to zero and solve for x.
Substitute x back into the line equation to find y.
Post you answer for checking if you wish.