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 =
No, I think the way to do it is to determine the line perpendicular to the given line.
y=-6x+9
line perpendicular to this is
y=1/6 x + b. Now it includes (-3,1)
1=1/6*-3 + b so b is 3/2..check that.
Now, find the original point..
y=1/6x+3/2 and
y=-6x+9
solve these two, to get x,y.
To find the point on the line 6x + y = 9 that is closest to the point (-3,1), we need to minimize the distance between this point and the given point (-3,1).
The distance d between two points (x1, y1) and (x2, y2) can be calculated using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, we want to minimize the distance d, so we can rewrite the distance formula as a function of x and y:
d = sqrt((x - (-3))^2 + (y - 1)^2)
Now we substitute the equation of the line 6x + y = 9 into the above equation to eliminate any variable:
d = sqrt((x - (-3))^2 + ((9 - 6x) - 1)^2)
Simplifying further:
d = sqrt((x + 3)^2 + (8 - 6x)^2)
Now we can find the minimum value of d by minimizing this function. One way to do this is by taking the derivative of d with respect to x, setting it equal to zero, and solving for x. However, this function is quite complex, and taking the derivative might not be the most efficient method.
Alternatively, we can use a geometric approach. Since we want to minimize the distance d, we want the point (x, y) to be perpendicular to the line.
The given line is in the form Ax + By = C, where A = 6, B = 1, and C = 9. The perpendicular line will have coefficients -B and A, so its equation will be -x + 6y = D, where D is a constant.
Now we substitute (-3,1) into this equation to find the value of D:
-(-3) + 6(1) = D
3 + 6 = D
D = 9
Therefore, the equation of the perpendicular line is -x + 6y = 9.
Now we solve the system of equations formed by the given line 6x + y = 9 and the perpendicular line -x + 6y = 9 to find the point of intersection, which will be the point on the given line closest to the point (-3,1).
6x + y = 9
-x + 6y = 9
Multiplying the second equation by 6:
6x + y = 9
-6x + 36y = 54
Adding the two equations together, we eliminate x:
37y = 63
y = 63/37
Now substitute this value of y into one of the original equations to solve for x:
6x + (63/37) = 9
6x = 9 - (63/37)
x = (9 - (63/37))/6
Therefore, the point on the line 6x + y = 9 that is closest to the point (-3,1) is (x, y) where x is (9 - (63/37))/6 and y is 63/37.