+864 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.
+129830 Slope of line = -1
Slope of any tangent line to the circle= -x/y
So
-x/y = -1
x/y =1
x = y
But
x + y =3 + k
x^2 + y^2 = k
So
x + x = 3+k
x^2 + x^2 = k
So
2x = 3+k → x = (3 + k) / 2
So
2x^2 = k
2 [ (3 + k) / 2] ^2 = k
(3 + k)^2 = 2k
k^2 + 6k + 9 = 2k
k^2 + 4k+ 9 = 0
The discriminant = 16 - 4 * 1 * 9 = -20
So....k is not a real value
So....no real solution
