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 = 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
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.
= x^2 + 6x + 9 + 64 - 96x + 36x^2
= 37x^2 - 90x + 73
To find the minimum value of D, we need to find the x-coordinate of the vertex of the quadratic function. The x-coordinate of the vertex is given by x = -b/2a, where a = 37 and b = -90.
x = -(-90) / (2 * 37)
x = 90 / 74
x = 1.2162 (rounded to 4 decimal places)
To find the y-coordinate of the vertex, we substitute the value of x into the equation for D:
D = 37(1.2162)^2 - 90(1.2162) + 73
D = 45.7751 (rounded to 4 decimal places)
Therefore, the minimum value of the distance squared D is approximately 45.7751. To find the point on the line that is closest to the point (-3,1), we substitute the value of x back into the equation for the line:
6(1.2162) + y = 9
7.2972 + y = 9
y = 9 - 7.2972
y = 1.7028 (rounded to 4 decimal places)
So, the point on the line 6x + y = 9 that is closest to the point (-3,1) is approximately (1.2162, 1.7028).
To find the point on the line 6x + y = 9 that is closest to the point (-3,1), we can follow these steps:
Step 1: Rewrite the equation of the line in terms of y:
6x + y = 9
y = 9 - 6x
Step 2: Substitute the equation y = 9 - 6x into the formula for d:
d = sqrt((x - (-3))^2 + (y - 1)^2)
d = sqrt((x + 3)^2 + (9 - 6x - 1)^2)
d = sqrt((x + 3)^2 + (8 - 6x)^2)
Step 3: To minimize the distance, we can minimize the square of the function D = d^2. So, we want to find the minimum of D:
D = (x + 3)^2 + (8 - 6x)^2
Now, to find the minimum point of D, we can take the derivative of D with respect to x and set it to zero:
dD/dx = 2(x + 3) + 2(8 - 6x)(-6)
0 = 2x + 6 - 12(8 - 6x)
0 = 2x + 6 - 96 + 72x
94 = 74x
x = 94/74
x = 1.27 (rounded to 2 decimal places)
Step 4: Substitute the value of x back into the equation for y to find the corresponding y-coordinate:
y = 9 - 6x
y = 9 - 6(1.27)
y = 1.62 (rounded to 2 decimal places)
Therefore, the point on the line 6x + y = 9 that is closest to the point (-3,1) is approximately (1.27, 1.62).