What is the locus of points with $x = 5\sec(\theta)$ and $y=4\tan(\theta)$?
I have this exercise, I think the answer is an hyperbola (I plotted in Wolfram), so maybe what is being asked is to transform these two equations in a more familiar equation of hyperbola. My attempt:
$$r\cos(\theta) = 5\sec(\theta) \\ r = 5\sec^2(\theta) \\\\r\sin(\theta)=4\tan(\theta)\\r=4\sec(\theta)\\4\sec(\theta)=5\sec^2(\theta)\\\sec(\theta)=\dfrac{4}{5}\\\theta=\text{arc}\sec\left(\dfrac{4}{5}\right)$$
But this is a line and does not make much sense...
Recall the formula $1+ \tan^2\theta = \sec^2\theta$, which means $\sec^2\theta-\tan^2\theta = 1$. Hence $$\left(\frac{x}{5}\right)^2 - \left(\frac{y}{4}\right)^2=1 \implies 16x^2-25y^2 = 400,$$which is the equation of a hyperbola.