## Rates of Change Geometry Problem

A point P is free to move along the

$x$
axis between points A and B.Line segments joint to point P to points R and S as shown, and the lines PR and PS make angles
$\theta$
and
$\phi$
with the positive and negative
$y$
axes respectively. How can you find
$\frac{D \theta}{D \phi}$
?

$tan \theta =\frac{AB-x}{b} \rightarrow sec^2 \theta \frac{d \theta}{dx}=- \frac{1}{b} \rightarrow \frac{d \theta}{dx}=- \frac{cos^2 \theta}{b}$

$tan \phi =\frac{x}{a} \rightarrow sec^2 \phi \frac{d \phi}{dx}= \frac{1}{a} \rightarrow \frac{d \phi}{dx}= \frac{cos^2 \phi}{a}$

Then
$\frac{d \theta}{d \phi}=\frac{\frac{d \theta}{dx}}{\frac{d \phi}{dx}}=\frac{- \frac{cos^2 \theta}{b}}{ \frac{cos^2 \phi}{a}}=- \frac{a cos^2 \theta}{b cos^2 \phi}$