geometry
posted by Lance on .
What are the equations of the lines through (5, 3) and passing at distance 2sqrt5 from (5,7)?
This problem is confusing? How do I solve this? Please I need your Help

all points 2√5 from (5,7) form a circle:
(x5)^2 + (y7)^2 = 20
Now you want a tangent to the circle that passes through (5,3). Must be a tangent, because any other line through the circle will come closer than 2√5 to the center.
So, since lines through (5,3) with slope m are
(y+3) = m(x+5), we need
(x5)^2 + (m(x+5)37)^2 = 20
(x5)^2 + (mx+(5m10))^2 = 20
(m^2+1)x^2 + 10m(m3)x + (5m10)^2  20 = 0
Now for the line to be tangent, the above equation must have a single root. That is, the discriminant must be zero:
100m^2(m3)^2  4(m^2+1)((5m10)^220) = 0
m = 1/2, 2
so, the two lines are
y = 1/2 (x+5)  3
y = 2(x+5)  3