SOLUTION: determine whether the given line is a tangent,secant or neither to the given circle in each case : x^2+y^2=9 and 2y-4x=6

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Question 1088655: determine whether the given line is a tangent,secant or neither to the given circle in each case :
x^2+y^2=9 and 2y-4x=6
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

We can "look and see" that the line is a secant for
it cuts the circle in two points.

But "look and see" doesn't count in rigorous mathematics,
so we solve the system to find whether there

1. are 2 real solutions, in which case it is a secant, as we suspect,
2. is only 1 real solution, in which case it would be a tangent,
3. are no real solutions (imaginary solutions), in which case
   it would be neither (a line totally outside the circle).

Solve the system of the two equations:

system%28x%5E2%2By%5E2=9%2C2y-4x=6%29

Solve the second for y

2y-4x=6
2y=4x%2B6
y=2x%2B3

Substitute for y in the first equation

x%5E2%2By%5E2=9
x%5E2%2B%282x%2B3%29%5E2=9
x%5E2%2B%284x%5E2%2B12x%2B9%29=9
x%5E2%2B4x%5E2%2B12x%2B9=9
5x%5E2%2B12x%2B9=9
5x%5E2%2B12x=0
x%285x%2B12%29=0

x=0;  5x+12=0
         5x=-12
          x=-2.4

If x = 0, then y = 2x+3 = 2(0)+3 = 3
So the line intersects the circle at (x,y) = (0,3)

If x = -2.4, then y = 2x+3 = 2(-2.4)+3 = -4.8+3 = -1.8
So the line intersects the circle also at (x,y) = (-2.4,-1.8)

So the line intersects the circle twice, so it is a secant.

Edwin