Seems like the image should have been
So, I have this curve C: y=2x^3 and a point P(p, 2p^3), p being a positive number. With the following graphic:
L1 and L2 are tangent lines of C passing through P.
I am asked to express the slope of L2 in terms of p.
Let be the slope of .
Then the equation of is: .
must intersect the graph in two distinct points. One point is and the other is the point where is tangent to the graph.
So, the system must have two solutions.
Replacing y from the second equation in the first equation we have
The last equation must have equal roots. Then
Here's a calc way if you'd still like to see it.
We can find where L2 is tangent to C by:
Solve this for x and we get
Sub this back into and we see
Now, we have two end points from which to find the slope of L2. Use the slope of a line formula.
Which agrees with red dogs clever method.