Grigori S. answered • 07/15/13

Certified Physics and Math Teacher G.S.

This is a standard form of a straight line. Write your equation in the slope - intercept form by subtracting the first term Ax from both sides of the equation and then dividng by B. You will obtain

y = -(A/B)x + C/B (1)

As you can see the slope "m"of this line is m = -(A/B). The slope "m_{1}" of the line perpendicular to (1) is defined as a reciprocal to "m" with negative sign: m_{1} = -1/m. Thus we have for the equation of the perpendicular line:

y = (B/A)x + c (2)

where "c" is to be found. In order to find "c" we have to use the point (a,b). That means, if x = a then y=b, or

b = (B/A) a + c (3)

Thus, c = b - (B/A)a, and equation (2) can be rewritten

y = (B/A)x + b - (B/A) a = (B/A)(x-a) + b (4)

Because the point of intersection belongs to both lines, we can make equal y-s of two (1 and 4). Thus we have

-(A/B) x + C/B = (B/A) x + b - (B/A)a (5)

Now solve (5) for x. You will obtain

x = [(C/B) +(B/A)a -b]/[(B/A) +(A/B)] (6)

Now you have left tom plug (6) into (1) to find y. It gives you

y = - (A/B)[(C/B) +(B/A)a - b]/[(B/A) + (A/B)] + C/B (7)

Formulas (6) and (7) give solution of the problem in a very generic form.

Grigori S.

Your solution is elegant. Well done!

07/15/13