Line AB has equation 4x+5y+2=0. If the point P=(p, p+5) lies on AB, find P . The point A has coordinates (1, 2). The point C(5, k) is such that AC is perpendicular to AB. Find the value of k.

i) Well, if point P lies in AB, then the value of its x and y coordinates have to fulfill the condition imposed by equation AB. Therefore, substituting the x value p for x and the y value p+5 for y:

4(p)+5(p+5)+2=0    Solving for p    p=-3    therefore     P=(-3,2)

ii) AC is perpendicular to AB. We know that if two lines are perpendicular the dot product between their respective direction vectors must be equal to 0. Therefore we start by calculating AC and AB:

AC=OC-OA=(5-1,k-2)=(4,k-2)

As for AB we know that the general equation of a 2D line is Ax+By+C=0, it direction vector being d=(-B,A). Hence, if AB is 4x+5y+2=0; AB=(-5,4).

Doing the dot product:

ACAB=4(-5)+(k-2)*4=0      solving for k   k=7   and therefore C=(5,7)