Tangent line for a parametric curve: parabola: For what values of a and b is the line

tangent to the parabola , when . From the equation of the line we can get its slope. Then we can use that slope and the derivative of the parabola to determine the parameter, a. First we need the line in slope-intercept form: From this we see that the slope is 2.

At the point of tangency, the slope of the parabola will have this value also.

The slope of the parabola at any point is .

We can equate these two slopes at :

, or

With 'a' determined, we can rewrite the equation of the parabola as .

In order to determine b in the equation of the tangent line, we need the coordinates of a point on the line. We can use the point of tangency for this purpose, obtaining its y-coordinate from the equation of the parabola:

. Hence the point is . Now we can use this point in the equation of the tangent line to get b: , or .

Finally we can substitute this into the general equation of the tangent line: