Convert the polar equation to a rectangular equation.
-$$r = \frac { 4 \sec \theta } { \sec \theta + 2 }$$
A) $3 x ^ { 2 } - y ^ { 2 } - 16 x + 16 = 0$
B) $3 y ^ { 2 } - x ^ { 2 } + 16 = 0$
C) $3 x ^ { 2 } - y ^ { 2 } + 16 = 0$
D) $3 y ^ { 2 } - x ^ { 2 } - 16 y + 16 = 0$

Question

Quizplus · Accepted Answer

To convert the given polar equation $r = \frac { 4 \sec \theta } { \sec \theta + 2 }$ to a rectangular equation, we use the relationships $r^2 = x^2 + y^2$, $x = r \cos \theta$, and $y = r \sin \theta$. The secant function is the reciprocal of the cosine function, so $\sec \theta = \frac{1}{\cos \theta}$. Substituting $r = x/\cos \theta$ into the given equation, we get $x/\cos \theta = \frac{4/\cos \theta}{1/\cos \theta + 2}$. Simplifying this equation and converting it into terms of $x$ and $y$ using the relationships above will lead to the rectangular equation $3 x ^ { 2 } - y ^ { 2 } - 16 x + 16 = 0$.

Convert the Polar Equation to a Rectangular Equation r=4sec⁡θsec⁡θ+2r = \frac { 4 \sec \theta } { \sec \theta + 2 }r=secθ+24secθ​

Convert the Polar Equation to a Rectangular Equation $r = \frac { 4 \sec \theta } { \sec \theta + 2 }$