Given a circle x2+y2=5 and a line y=2x+k find the values of k such that
1) the line touches the circle at exactly one point.
2) the line intersects the circle at some points.
3)the line does not cut the circle at all.
+129839 (1)
x^2 + y^2 = 5 (1)
y = 2x + k (2)
The slope of the line is 2......usiing implicit differentiation...the slope of any tangent line to the circle is given by .... -x / y
So ....equating slopes
-x / y = = 2 → x = -2y (3) ...... sub this into (1)
(-2y)^2 + y^2 = 5
4y^2 + y^2 = 5
5y^2 = 5 → y = ± 1
When y = ± 1 ....using (3), x = - 2 or x = 2
So....we have the points (-2, 1) and (2, - 1)
And using (2) and solving for k........ k = 5 or k = - 5
Here's a graph : https://www.desmos.com/calculator/qkhdev0shy
(2)
And it's obvious that the line will intersect the circle at two points whenever
-5 < k < 5
(3)
And the line will not intersect the circle whenever
k < - 5 or k > 5
