Tangents to a parabola through a given point: For the parabola , find the equations of the two tangent lines which pass through . Also find their points of tangency.

First we need the slope of the tangent line at any point: the derivative

.

This will be the slope of the tangent line, and -9 will be the y-intercept. Hence the equation of the tangent line
at the point of tangency is

.

Since this point is also on the parabola, it must satisfy the equation of the parabola:

.

We can equate these two results for to get

.

Solving ,

, or .

With these values of , we can find corresponding y coordinate from the equation of the parabola:

.

To summarize, the points of tangency of the two lines are and