Is x+y+1=0 a tangent of both y^2=4x and x^2=4y parabolas?

Accepted Solution

Answer:   yesStep-by-step explanation:The line intersects each parabola in one point, so is tangent to both.__For the first parabola, the point of intersection is ...   y^2 = 4(-y-1)   y^2 +4y +4 = 0   (y+2)^2 = 0   y = -2 . . . . . . . . one solution only   x = -(-2)-1 = 1The point of intersection is (1, -2).__For the second parabola, the equation is the same, but with x and y interchanged:   x^2 = 4(-x-1)   (x +2)^2 = 0   x = -2, y = 1 . . . . . one point of intersection only___If the line is not parallel to the axis of symmetry, it is tangent if there is only one point of intersection. Here the line x+y+1=0 is tangent to both y^2=4x and x^2=4y._____Another way to consider this is to look at the two parabolas as mirror images of each other across the line y=x. The given line is perpendicular to that line of reflection, so if it is tangent to one parabola, it is tangent to both.