1. You have the parabolas:
and
2. Let P(p, p²) denote the tangent point on the parabola p and Q(q, -q²+6q-5) the tangent point on the parabola q.
3. The tangent to p at P has the equation:
and the tangent to q at Q has the equation
4. Both equations describe the same line. Thus
5. Solve this system of simultaneous equations for p and q. Resubstitute the result into the equations of and .
6. For confirmation: Draw the 2 parabolas and the tangents.
Here's a tip- in the future tell us what the problem really says! Here, you have never said exactly which tangent lines you want to find. Every graph has an infinite number of tangent lines.
I suspect that the problem is asking for lines that at tangent to both of these graphs. Suppose y= mx+ b is the equation of such a tangent line. At the point, where that line is tangent to we must have and . At the point, where it is tangent to , we must have and so we have 4 equations to solve for m, b, , and