# Find the point on the line y=6x+9 that is closest to the point (-3,1). Clearly state the domain of the function that you optimize. (Instead of minimizing the distance d, you can minimize d^2 as it...

Find the point on the line y=6x+9 that is closest to the point (-3,1). Clearly state the domain of the function that you optimize. (Instead of minimizing the distance *d*, you can minimize *d^2* as it is easier.)

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### 1 Answer

The distance between (x,y) on the line and (-3,1) is as follows:

`d^2=(x-(-3))^2+(y-1)^2=(x+3)^2+(y-1)^2`

Substitute y=6x+9 into the equation:

`d(x)^2=x^2+6x+9+(6x+9-1)^2`

`= x^2+6x+9+36x^2+96x+64`

`= 37x^2+102x+73`

Find the minimum value of x for which the derivative is 0:

`(d(d(x)^2))/(dx)=74x+102=0 -gtx=-1.38`

f(-1.38)=6(-1.38)+9=0.72

Therefore the point closest to (-3,1) on the line y=6x+9 is (-1.38,0.72)

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