+859 Let $k$ be a positive real number. The line $x + y = 3 + k$ and the circle $x^2 + y^2 = k$ are drawn. Find $k$ so that the line is tangent to the circle.
+129881 x + y = 3 + k
y = -x + (3 + k)
The slope of the line = -1
The slope of a tangent line at any point on the circle = -x/y
Set the slopes equal
-x/y = -1
-x= -y
x = y
So....subbing into both equations
x + x = 3 + k → 2x = 3 + k → x = (3 + k)/2 (1)
x^2 + x^2 = k → 2x^2 = k (2)
Sub (1) into (2)
2 [ (3 + k)/2]^2 = k simplify
k^2 + 6k + 9 = 2k
k^2 + 4k + 9 = 0
k cannot be a real number because the discriminant = -20
