Write the equation of the tangent line to the curve x^2/8 - y^2/4 =1 at the point (4,2) by using the following facts. The slope m of the tangent line to a hyperbola at the point (x, y) is: m=b^2x/a^2y for x^2/a^2 -y^2/b^2=1 m=a^2x/b^2y for y^2/a^2 - x^2/b^2 =1

Step-by-step explanation:

So we are given the formula for the slope of a hyperbola in this form:

.

That formula for the slope is

If we compare the following two equations, we will be able to find and :

We see that while .

So the slope at is:

.

Recall: Slope-intercept form of a linear equation is .

We just found . Let's plug that in.

To find , the -intercept, we will need to use a point on our tangent line. We know that it is going through .

Let's enter this point in to find .

Subtract 4 on both sides:

Simplify:

The equation for the tangent line at on the given equation is: