Conversion of parametric equation to Cartesian form

Question

Can anyone help me convert this equation from parametric form to its Cartesian form?

$$ x = 7(\sec (t) + \tan (t)) $$

$$ y = 7(\sec (t) - \tan(t)) $$

Answer 1

You can compute $$ x+y=14\sec(t)\text{ and }x-y=14\tan(t). $$ Consider $\sec^2(t)=1+\tan^2(t)$. Therefore you write $$ x-y=14\tan(t)\Leftrightarrow \frac1{14^2}(x-y)^2=\tan^2(t) $$ and combine it with the first equation $$ (x+y)^2=14^2\sec^2(t)=14^2(1+\tan^2(t))=14^2\left(1+\frac1{14^2}(x-y)^2\right)=14^2+(x-y)^2 $$ Expand both squares and you get $$ x^2+2xy+y^2=14^2+x^2-2xy+y^2\Leftrightarrow 4xy=14^2\Leftrightarrow xy=49. $$ The answer of shmop is much shorter. But this might give an alternative idea how to deal with the equations.

Answer 2

Simply multiply the two equations together: $$xy = 49(\sec^2t-\tan^2t)=49.$$