Question

Whar is the distance from the origin to the line x/a + y/b = 1?

Answer 1

Putting your equation into slope-intercept form, we get:  

y = (-b/a)x + b  

So its slope is (-b/a), meaning the slope of a perpendicular line must be (a/b). Since we want the perpendicular line to go through the origin it's y-intercept is 0, and we write the perpendicular line as:  

y = (a/b)x  

Now to find the point of intersection, use substitution to get:  

(a/b)x = (-b/a)x + b, which simplifies to  

(a/b + b/a)x = b  

((a^2+b^2)/(ab))x = b  

So x = (a*b^2)/(a^2+b^2)  

To find the y coordinate, we substitute into our perpendicular line (the easier of the two)  

Thus y = (a/b)x = (a/b)*((a*b^2)/(a^2+b^2)) = (b*a^2)/(a^2+b^2)  

So the point closest to the origin is:  

(x,y) = ( (a*b^2)/(a^2+b^2), (b*a^2)/(a^2+b^2) )