How many times does this line intersect the given circle?
Center of the circle is at x=-2 and y=1. Line equation can be rewritten as y=-x/2 – 3/2, so its slope is -1/2. A perpendicular to this line would have slope +2 because the product of slopes of perpendicular lines is -1. So the equation of the perpendicular line going through the center of the circle is y=2x+C where C is some constant. Plugging in x=-2 and y=1 we see that C=5. Therefore the original line y=-x/2 – 3/2 and its perpendicular that runs through the center of the circle y=2x+5 will intersect at x=-13/5 and y=-1/5. The distance from the center of the circle to this point is sqrt((-2+13/5)^2 + (1+1/5)^2) = 3/sqrt(5). The radius of the circle is sqrt(6) is larger than this distance so there must be two points of intersection between the original line and the circle (if it were equal there would be one point = “tangent” and if were smaller there would be no intersection at all).
