Question

Find the point on the line 3 x + 2 y - 5 =0 which is closest to the point (4,5).

Answer 1

Take any point on the line as (x,y)

Now, distance between the point on the line and the given point = d

=> d^2 = (x-4)^2 + (y-5)^2

You can treat this as a function. When d^2 is minimum, then d will be minimum too.

function to minimize = (x-4)^2 + (y-5)^2

But, y can be written in terms of x:
3x + 2y = 5
=> y = (5-3x)/2

So, by minimizing the function
f(x) = (x-4)^2 + ((-3x-5)/2)^2
You will get the required point.