Find an equation of the tangent line to the graph of the function f defined by the following equation at the indicated point.

Question

Find an equation of the tangent line to the graph of the function f defined by the following equation at the indicated point.

y^2-x^2=486; (9,9sqrt7)

y= ???

Answer 1

Recall that the slope of the tangent line is given by the derivative, dy/dx, at the given point (9,9√7).

So we first need to use implicit differentiation on the equation y² - x² = 486, to find dy/dx:

2y·(dy/dx) - 2x = 0

We solve for dy/dx:

dy/dx = (2x)/(2y) = x/y.

To find the slope of the tangent line at the point (9,9√7), we substitute x = 9 and y = 9√7 into the formula that we just found for dy/dx.

So the slope of the tangent line is 9/(9√7) = 1/√7. If you want, you can rationalize the denominator so that the slope looks like (√7)/7.

Now you just need to write the equation of the line whose slope is 1/√7 (or (√7)/7) and that goes through the point (9,9√7). This last step is something that you should be familiar with.