Problem 74

Given information

The given parabolas are :

We are required to find if there is a line that is tangent to both lines.

Step-by-step explanation

The parabola intersect the -axis at .

Notice that is a quadratic formula with a negative discriminant, therefore it has no real solution.

All points in the parabola has the form .

All points on the parabola has the form .

We know that the slope of the tangent at any point is the derivative at the point : .

Applying the Power Rule and the Constant Multiple Rule, we obtain :

Assume that there is a tangent that passes through those two points.

Hence, the required derivative has common solution: