Question

find the ranger of values of m such as the straight line y = 2x + m intersects the curve x^2 + y^2 = 4 at two distinct points

Answer 1

typo there range*

Answer 2

x^2 + y^2 = 4
is a circle with radius 2, centered at the origin

find the y-intercepts at those 2 tangent points to get range of m

dy/dx = slope of tangent line = 2

Answer 3

Put y = 2x + m into x^2 + y^2 = 4
x^2 + (2x + m)^2 = 4
5x^2 + 4mx +m^2-4 =0
For it to have 2 distinct intersection points, Δ>0
(4m)^2 - 4(5)(m^2-4) >0
-4m^2 +80>0
4m^2<80
m^2 < 20
-sqrt20<m<sqrt20