## Calculus (3rd Edition)

$r = \frac{4}{{7\cos \theta - \sin \theta }}$ is the polar equation of the line $7x - y = 4$
Since $r = \frac{4}{{7\cos \theta - \sin \theta }}$, we have ${r^2} = \frac{{16}}{{49{{\cos }^2}\theta - 14\cos \theta \sin \theta + {{\sin }^2}\theta }}$ Since $x = r\cos \theta$ and $y = r\sin \theta$, we get ${x^2} + {y^2} = {r^2}$. $\cos \theta = \frac{x}{{\sqrt {{x^2} + {y^2}} }}$, ${\ \ \ }$ $\sin \theta = \frac{y}{{\sqrt {{x^2} + {y^2}} }}$. So, ${x^2} + {y^2} = \frac{{16}}{{49{x^2}/\left( {{x^2} + {y^2}} \right) - 14xy/\left( {{x^2} + {y^2}} \right) + {y^2}/\left( {{x^2} + {y^2}} \right)}}$ ${x^2} + {y^2} = \frac{{16\left( {{x^2} + {y^2}} \right)}}{{49{x^2} - 14xy + {y^2}}}$ ${x^2} + {y^2} = \frac{{16\left( {{x^2} + {y^2}} \right)}}{{{{\left( {7x - y} \right)}^2}}}$ ${\left( {7x - y} \right)^2} = 16$ $7x - y = 4$ This is the equation of a line. Hence, $r = \frac{4}{{7\cos \theta - \sin \theta }}$ is the polar equation of the line $7x - y = 4$.