
Limits review problem
I'm not sure how to go about solving this problem. = \
Could someone please give me some suggestions as to how to begin.
"Find the equations of both lines that pass through (4,10) and are tangent to the parabola y = x^2 10x + 18."
Many thanks in advance,
dMh

Lets go back to point slope form: (I don't know subscripts so y1 = b and x1 = a)
$\displaystyle yb=m(xa) $
Now, we know that they must pass through 4, 10. Thus, a = 4, b = 10
We also know that the other point must be at (x,y). And we know the equation for y in terms of x.
We also know that in order to be tangent, the slope must equal the slope of the parabola. Luckily, we have derivatives to generalize this. The derivative of the parabola is simply 2x10. This is the m value.
$\displaystyle x^210x+18+10=(2x10)(x4)$
Now, solving this equation for x, we get x = 2,6.
Therefore, we have lines that go through (4,10) to (2,2) and (4,10) to (6,6).
Use point slope form to calculate the equations of the lines, and you're done.