Find the point on the line -6 x + 5 y - 6 =0 which is closest to the point ( 4, 5 ).

Question

 please help answer asap

Answer 1

5y = 6x + 6

 

y = (6/5)x + (6/5)

 

 

The point (x , (6/5)x + (6/5)) is closer to (4, 5).

 

d = √[(x - 4)² + ((6/5)x + (6/5) - 5)²

 

d = √[(x² - 8x + 16) + ((6/5)x - (19/5))²]

 

d = √[(x² - 8x + 15) + (1.44x² - 9.12x + 14.44)]

 

d = √(2.44x² - 17.12x - 29.44)

 

 

Take the derivative of this and set it equal to zero.

 

4.88x - 17.12 = 0

 

 

Solving for x,

 

4.88x = 17.12

 

      x = 214/68

 

 

This confirms that Christopher's work is correct.   If the program you are using says it is wrong, then try using entering decimal numbers rather than the fractional answer.

Answer 2

Another way to solve this, without calculus:

 

Find the equation of the line perpendicular to the given line, and passing through (4,5).  Its slope must be 

 

m = -1/(6/5) = -5/6

 

y = mx + b = (-5/6)x + b

 

Find b from the given point.

 

5 = -(5/6)(4) + b

b = ?

 

You will have the equation of a line perpendicular to the given line and passing through (4,5).  The intersection of the given line and the perpendicular line is the point on the given line that is closest to (4,5).  Just solve the system of 2 linear equations in x, y to get the intersection.

Answer 3

So here you want to minimize the distance between the point (4,5) and some point (x,y) that exists on the line. We know that the distance formula is D = sqrt((x-4)^2 + (y-5)^2), from this relationship and the equation of the line above we can get d to be a function of either x alone or y alone. from there we take the derivative and set it equal to zero to find the critical points. This is done easiest using implicit differentiation:

2Dd(D)/dx = 2(x-4) + 2(y-5)dy/dx, solve for d(D)/dx

d(D)/dx = ((x-4) + (y-5)dy/dx)/D, and set this to zero

(x-4) + (y-5)dy/dx = 0, note that I didn't divide by D since I can just multiply it out once we've set the equation to zero

Now plug in y and dy/dx and solve for x

x-4 + (36/25)x + (36/25) - 6 = 0 , here dy/dx = 6/5, and y = (6/5)(x + 1)

solving this for x we get: x = 214/61

Then plug this back into the y equation and you will get the point on the line which is closest