Question

Find an equation of the tangent line to the hyperbola y = 3/x at the point (3, 1).

Answer 1

Let's find the derivative of that hyperbola in order to find a slope formula to help us with this equation.  We already have an x and y value.  The derivative is found this way:  $y'= \frac{x(0)-3(1)}{x^2}$ so  $y'=- \frac{3}{x^2}$.  The derivative supplies us with the slope formula we need to write the equation.  Sub in the x value of 3 to find what the slope is: $y'=- \frac{3}{3^2}=- \frac{3}{9}=- \frac{1}{3}$.  So in our slope-intercept equation, x = 3, y = 1, and m = -1/3.  Use these values to solve for b.  $1=- \frac{1}{3}(3)+b$  so b = 2.  The equation, then, for the line tangent to that hyperbola at that given point is $y=- \frac{1}{3}x+2$