While finding a formula for the distance from a point to a l

While finding a formula for the distance from a point to a line, we had to solve the system of linear equations {y = mx + c y = -1/m(x - a) + b. Find the intersection point of these two lines.

Solution

From the given linear equations,

y= mx + c = (-1/m)(x-a) + b

or mx + c = (-1/m)x + (a/m) + b

or [m+(1/m)]x = a/m + b - c

or x = [a + bm - cm] / [m² + 1]

from the first equation,

y = mx + c

or y = ( m [a + bm - cm] / [m² + 1] ) + c

or y = [am + bm² - cm² + cm² + c] / [m² + 1]

or y = [am + bm² + c] / [m² + 1]

intersection point = (x,y) = ( [a + bm - cm] / [m² + 1] , [am + bm² + c] / [m² + 1] )