The polar equation of the conic with the given eccentricity and directrix and focus at origin: r=41 + cos theta

Step 1 If the conic of directrix ${\displaystyle x=\pm p}$ where p is positive real number and essentricity is e, and focus at origin then the polar equation of conic is: ${\displaystyle r=ep\pm e\cos \theta }$

Step 2 It is provided that eccentriciti is ${\displaystyle e=1}$ and directrix is ${\displaystyle x=4}$ and focus at the origin. To find the polar equation of conic put the provided value in above equation as: ${\displaystyle r=1\times 41\pm 1\times \cos \theta }$ Since, here directrix is positive ${\displaystyle x=4}$ so, consider the positive sign as: ${\displaystyle r=41+\cos \theta }$ Therefore, polar equation is ${\displaystyle r=41+\cos \theta }$