A circle with radius 4 and center point (1,c) is tangent to the line y=5x-2. Find all possible solutions for c.
I tried solving this but I only got one solution...help?
I tried solving this but I only got one solution...help?
-
Equation of circle is (x-1)^2 + (y -c)^2 = 16
If y = 5x-2 is tangent to the circle then system
(x-1)^2 + (5x- 2 - c)^2 =16
y = 5x-2
has only one solution else the line y = 5x-3 cuts the circle more than one point, Δ must be 0 for this
(x-1)^2 + (5x- 2 - c)^2 =16 or
26x^2 + x ( -2 - 20 - 10c) + 1 + (c+2)^2 - 16 = 0
Δ = (10c + 22)^2 - 104[(c+2)^2 -15] = 100c^2 + 440c+ 484 - 104c^2 - 416c + 1144 = 0
- 4c^2 + 24c + 1584 = 0
c^2 - 6c - 396 = 0
Solve for c and find two value of c
c = 3 ± √405 = 3 ± 9√5
If y = 5x-2 is tangent to the circle then system
(x-1)^2 + (5x- 2 - c)^2 =16
y = 5x-2
has only one solution else the line y = 5x-3 cuts the circle more than one point, Δ must be 0 for this
(x-1)^2 + (5x- 2 - c)^2 =16 or
26x^2 + x ( -2 - 20 - 10c) + 1 + (c+2)^2 - 16 = 0
Δ = (10c + 22)^2 - 104[(c+2)^2 -15] = 100c^2 + 440c+ 484 - 104c^2 - 416c + 1144 = 0
- 4c^2 + 24c + 1584 = 0
c^2 - 6c - 396 = 0
Solve for c and find two value of c
c = 3 ± √405 = 3 ± 9√5