Convert the polar equation to rectangular form. $r = \frac { 8 } { 1 + \sin \theta }$
A) $x ^ { 2 } + 16 y - 64 = 0$
B) $x ^ { 2 } + 16 x + 64 = 0$
C) $x ^ { 2 } + 16 y - 64 x = 0$
D) $x ^ { 2 } + 16 y + 64 = 0$
E) $x ^ { 2 } + 16 - 64 = 0$

Question

Convert the polar equation to rectangular form.  $r = \frac { 8 } { 1 + \sin \theta }$
A) $x ^ { 2 } + 16 y - 64 = 0$
B) $x ^ { 2 } + 16 x + 64 = 0$
C) $x ^ { 2 } + 16 y - 64 x = 0$
D) $x ^ { 2 } + 16 y + 64 = 0$
E) $x ^ { 2 } + 16 - 64 = 0$

Quizplus · Accepted Answer

To convert the given polar equation $r = \frac { 8 } { 1 + \sin \theta }$ to rectangular form, we use the relationships $r^2 = x^2 + y^2$, $r\cos\theta = x$, and $r\sin\theta = y$. The given equation can be rewritten as $r(1 + \sin\theta) = 8$, which simplifies to $r + r\sin\theta = 8$. Substituting the rectangular form equivalents, we get $r + y = 8$, and since $r^2 = x^2 + y^2$, we substitute $r = 8 - y$ into $r^2 = x^2 + y^2$ to get $(8 - y)^2 = x^2 + y^2$. Expanding and simplifying leads to the equation $x^2 + 16y - 64 = 0$, which matches choice A.

Convert the Polar Equation to Rectangular Form r=81+sin⁡θr = \frac { 8 } { 1 + \sin \theta }r=1+sinθ8​

Convert the Polar Equation to Rectangular Form $r = \frac { 8 } { 1 + \sin \theta }$