Find the point on the line 6𝑥+7𝑦−3=0 which is closest to the point (2,3).

Answer 1

The equation is rewritten as y = 3/7 - (6/7)x

Since we want to minimize distance, we employ the distance equation

D = sqrt{ (y-y₁)² + (x-x₁)² } where (x₁,y₁) is the point (2,3)

Because square roots are cumbersome to deal with, we can minimize D² just the same to get the correct answer. Remember that y = 3/7 - (6/7)x.

So, let D² = { (3/7-(6/7)x -3)² + (x-2)² } = 520/49 + (20/49)x + (85/49)x²

then,

dD²/dx = 20/49 + (170/49)x Setting this equal to zero, we get x = -(2/17)

Substituting back into the equation for y, we get y = 63/119 = 9/17

So, the closest point on the line closest to (2,3) is (-2/17, 9/17) (If I did all the math correctly?)