Answer
$r = \frac{4}{{7\cos \theta - \sin \theta }}$ is the polar equation of the line $7x - y = 4$
Work Step by Step
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 $7x - y = 4$.
