# Math Help - [SOLVED] Calculus Word Questions

1. ## [SOLVED] Calculus Word Questions

2. Hello, CAIR!

Find the equation of the tangent to the hyperbola: . $\frac{x^2}{16} - \frac{y^2}{9} \:=\:1$
at the point $\left(\text{-}5,\tfrac{9}{4}\right)$ on the curve.
We have: . $9x^2 - 16y^2 \:=\:144$

Differentiate implicitly: . $18x - 32y\frac{dy}{dx} \:=\:0 \quad\Rightarrow\quad \frac{dy}{dx} \:=\:\frac{9x}{16y}$

At $\left(\text{-}5,\tfrac{9}{4}\right),\;\;\frac{dy}{dx} \:=\:\frac{9(\text{-}5)}{16(\frac{9}{4})} \:=\;-\frac{5}{4}$ . . . slope of the tangent

The equation of the line through $\left(\text{-}5,\tfrac{9}{4}\right)$ with slope $m = \text{-}\tfrac{5}{4}$ is:

. . $y - \tfrac{9}{4} \;=\;\text{-}\tfrac{5}{4}(x + 5) \quad\Rightarrow\quad\boxed{ y \:=\:\text{-}\tfrac{5}{4}x - 4}$

3. thanks