I need help with a linear equations question

Question

Hi I am currently having issues solving a question relating to linear equations: A family of lines have equations of the form y=-2x+c, where c is a positive number.

Let O denote the origin, and let A and B be the x-axis and y-axis intercepts of y=-2x+c.

If the midpoint M of AB is such that OM=2 x sqrt5, find the value of c.

Answer 1

c=8

helps to draw a sketch, if only a rough one

OM = 2sqr5, where M = Midpoint between the x and y intercepts of y= -2x +c

slope = -2, c = y intercept = A = (0,a) = (0, c)

y= -2x +c

2x = -y +c

x = -y/2 +c/2, c/2 = x intercept = B = (b, 0) = (c/2, 0)

midpoint of A and B is the average of their coordinates = (c/4, c/2) = M

distance d from O to M satisfies

d^2 = c^2/16 +c^2/4 = (c^2 +4c^2)/16 = 5c^2/16

d = sqrd^2 = csqr5/4 = 2sqr5

c/4 = 2

c= 8

y = -2x + 8 with y intercept = 8, x intercept = 4

A = (0,8), B = (4,0)

M = (2,4)

OM = sqr(4+16) =sqr20 = 2sqr5