Find the eccentricity of the conic. $r = \frac { 7 } { 8 - 7 \sin \theta }$
A) $e = \frac { 8 } { 7 }$
B) $e = 7$
C) $e = - 7$
D) $e = \frac { 7 } { 8 }$
E) $e = 8$

Question

Quizplus · Accepted Answer

For a conic given in polar coordinates, the eccentricity can be found using the formula $e=\frac{r_{max}}{r_{min}}-1$, where $r_{max}$ and $r_{min}$ are the maximum and minimum distances from the focus to a point on the conic. For this conic, $r_{max}=8$ (when $\sin 	heta = 1$) and $r_{min}=1$ (when $\sin 	heta = -1$). Thus, $e=\frac{8}{1}-1=7$. However, the multiple choice answers are given in terms of the semi-latus rectum $p$, which is related to the eccentricity by $p=\frac{b^2}{a}=a(1-e^2)$. We can find $a$ and $b$ by converting to rectangular coordinates:
\begin{align*}
r&=\frac{7}{8-7\sin 	heta} \
x&= r \cos 	heta = \frac{7 \cos 	heta}{8-7\sin 	heta} \
y&= r \sin 	heta = \frac{7\sin 	heta}{8-7\sin 	heta}
\end{align*}
Squaring and adding $x$ and $y$ gives
\begin{align*}
x^2+y^2 &= \frac{49}{(8-7\sin 	heta)^2} \
49(8-7\sin 	heta)^2 &= (x^2+y^2) 8^2 \
49(8-7\sin 	heta)^2 &= (x^2+y^2) (x^2+y^2+49) \
7(8-7\sin 	heta) &= \sqrt{(x^2+y^2)(x^2+y^2+49)} \
7(8-7\sin 	heta) &= \sqrt{(r^2)(r^2+49)} \
7(8-7\sin 	heta) &= \sqrt{\left(\frac{7}{8-7\sin 	heta}ight)^2\left(\frac{49}{(8-7\sin 	heta)^2}+49ight)} \
7(8-7\sin 	heta) &= \sqrt{\frac{49}{8-7\sin 	heta}+49}
\end{align*}
Squaring both sides and simplifying gives the equation of the conic in rectangular coordinates:
$$49x^2-98xy+49y^2+392x-343y=0$$
To find the semi-latus rectum $p$, we need to convert this equation to standard form. Completing the square gives:
$$49\left(x-\frac{7}{2}ight)^2+49\left(y-\frac{4}{7}ight)^2=\frac{2401}{196}$$
Dividing both sides by $\frac{2401}{196}$ gives:
$$\frac{(x-7/2)^2}{49/2401} + \frac{(y-4/7)^2}{49/2401} = 1$$
Thus, $a=49/49=1$ and $b=\sqrt{2401/49-1}=7\sqrt{3}/7=\sqrt{3}$, so $p=b^2/a=\boxed{3}$.

Find the Eccentricity of the Conic r=78−7sin⁡θr = \frac { 7 } { 8 - 7 \sin \theta }r=8−7sinθ7​

Find the Eccentricity of the Conic $r = \frac { 7 } { 8 - 7 \sin \theta }$