Question

Q5: Determine the graph of the polar equation r = 10/10-7cos theta.

Answer 1

$r=\dfrac{10}{10-7\cos\theta}\implies10r-7r\cos\theta=10$

Converting to Cartesian coordinates using

$x=r\cos\theta$

$r=\sqrt{x^2+y^2}$

we have

$10\sqrt{x^2+y^2}-7x=10$

$10\sqrt{x^2+y^2}=7x+10$

$100(x^2+y^2)=(7x+10)^2$

$100x^2+100y^2=49x^2+140x+100$

$51x^2-140x+100y^2=100$

Complete both squares to get

$51\left(x-\dfrac{70}{51}\right)^2-\dfrac{4900}{51}+100y^2=100$

$51\left(x-\dfrac{70}{51}\right)^2+100y^2=\dfrac{10,000}{51}$

$\dfrac{2601}{10,000}\left(x-\dfrac{70}{51}\right)^2+\dfrac{51}{100}y^2=1$

This is the equation of an ellipse, so it's either A or C.

The ellipse is centered at $\left(\dfrac{70}{51},0\right)\approx(1.37,0)$, so it falls to the right of the vertical axis, which means the answer is C.