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

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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Let a point on 6x+y = 9 be (a,b)

The line formed by (a,b) and (-3,1) will be purpendicular to 6x+y = 9 when (a,b) is the closest point.

Equation of the line between (a,b), (-3,1)

(y-b)/(x-a) = (y-1)/(x+3)

      y(3+a) = x(b-1)+3b+a

             y  =(b-1)/(3+a)*x+(3b+a)/(3+a)

 

Since the lines are perpendicular the multiplication of thier gradients will equal -1.

 

y=-6x+9

y  =(b-1)/(3+a)*x+(3b+a)/(3+a)

 

-6*(b-1)/(3+a) =-1

6(b-1) = (3+a)

6b-a = 9------(1)

 

Also (a,b) is on the line 6x+y = 9

Then;

6a+b = 9-----(2)

 

solving (1) and (2) will give;

a=45/37 and b= 63/37

 

So the closest point is (45/37,63/37)

 

 

Sources:

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