First, we need to determine the slope from the two points given in the problem.
The slope can be found by using the formula: #m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#
Where #m# is the slope and (#color(blue)(x_1, y_1)#) and (#color(red)(x_2, y_2)#) are the two points on the line.
Substituting the values from the problem gives:
#m = (color(red)(2q) - color(blue)(q))/(color(red)(-p) - color(blue)(p))#
#m = q/-2p = -q/(2p)#
Now we can use either point to build a equation in the point-slope form:
The point-slope formula states: #(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))#
Where #color(blue)(m)# is the slope and #color(red)(((x_1, y_1)))# is a point the line passes through.
Solution for point (p, q):
#(y - color(red)(q)) = color(blue)(-q/(2p))(x - color(red)(p))#
Solution for point (-p, 2q):
#(y - color(red)(2q)) = color(blue)(-q/(2p))(x - color(red)(-p))#
#(y - color(red)(2q)) = color(blue)(-q/(2p))(x + color(red)(p))#
