Question

question
convert the given polar equation into a cartesian equation.
r = (sin θ + 7 cos θ)/(cos² θ - sin² θ)

Explanation:

Step1: Recall the relationships

We know that $x = r\cos\theta$, $y = r\sin\theta$, and $r^{2}=x^{2}+y^{2}$. First, multiply both sides of the equation $r=\frac{\sin\theta + 7\cos\theta}{\cos^{2}\theta-\sin^{2}\theta}$ by $r(\cos^{2}\theta - \sin^{2}\theta)$ to get $r^{2}(\cos^{2}\theta-\sin^{2}\theta)=r\sin\theta + 7r\cos\theta$.

Step2: Substitute $x$ and $y$

Since $\cos^{2}\theta-\sin^{2}\theta=\frac{x^{2}-y^{2}}{r^{2}}$ (because $\cos\theta=\frac{x}{r}$ and $\sin\theta=\frac{y}{r}$), $r\sin\theta = y$ and $r\cos\theta=x$, the left - hand side is $(x^{2}+y^{2})\frac{x^{2}-y^{2}}{r^{2}}\times r^{2}=x^{4}-y^{4}$, and the right - hand side is $y + 7x$. So the Cartesian equation is $x^{4}-y^{4}=y + 7x$.

Answer:

$x^{4}-y^{4}=y + 7x$