need help with limits and secants

Question

The solutions (x,y) of the equation x^2 + 16y^2 = 16 form an ellipse circle Consider the point P ( in the positive x-yplane), with x-coordinate 1.
a) Let h be a small non-zero number and form the point Q with x-coordinate 1+h, . The slope of the secant line through PQ, denoted s(h), is given by the formula?
b) Rationalize the numerator of your formula in (a) to rewrite the expression so that it looks like f(h)/g(h), subject to these two conditions: (1) the numerator f(h) defines a line of slope -1, (2) the function f(h)/g(h) is defined for h=0.
(c) The slope of the tangent line to the ellipse at the point P is?

I am able to get the answer to a which is (sqrt((15-2h-h^2)/16)-sqrt(15/16))/1+h-1, i am having trouble rationalizing the numerator.

Answer 1

Good work!

To rationalize the numerator of: use the difference of square formula.

Which simplifies to: (16−(1+h)^2)−15.

Simplify further, and the rationalized numerator is: 1-2h-2h^2

The final rationalized form:

s(h)=1−2h−h^2​/4h⋅(sqrt16−(1+h)^2​)+(sqrt15)​