If sum of the perpendicular distances of a variable point P (x, y) from the lines x + y – 5 = 0 and 3x – 2y +7 = 0 is always 10. Show that P must move on a line.

The equations of the given lines are:

x + y – 5 = 0 ---------------- (1)

3x – 2y +7 = 0 ------------ (2)

The perpendicular distance of P(x, y) from line (1) is given by

and

i.e, and

It is given that

Thus,

(Assuming (x+y-5) and (3x-2y+7))

Which is the equation of the line.

Similarly, we can get the equation of line for any signs of (x + y -5) and (3x – 2y + 7).

Therefore, point P must move on a line.