We realize that the directions of a point separating the line portion joining the focuses (x1, y1) and (x2, y2) inside in the proportion m: n are
We know that slope, m = (y2 – y1)/(x2 – x1)
= (3 – 0)/(2 – 1)
= 3/1
= 3
We know that two non-vertical lines are perpendicular to each other if and only if their slopes are negative reciprocals of each other.
Then, m = (-1/m) = -1/3
We know that the point (x, y) lies on the line with slope m through the fixed point (x0, y0), if and only if, its coordinates satisfy the equation y – y0 = m (x – x0)
Here, the point is
3((1 + n) y – 3) = (-(1 + n) x + 2 + n)
3(1 + n) y – 9 = – (1 + n) x + 2 + n
(1 + n) x + 3(1 + n) y – n – 9 – 2 = 0
(1 + n) x + 3(1 + n) y – n – 11 = 0
∴ The equation of the line is (1 + n) x + 3(1 + n) y – n – 11 = 0.