Write a polar equation in $r$ and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $0.8$ and vertex at $\left( 1 , \frac { \pi } { 2 } \right)$.
A) $r = \frac { 1.8 } { 1 - 4 \cos \theta }$
B) $r = \frac { 9 } { 1 + 4 \sin \theta }$
C)
$r = \frac { 1.8 } { 5 - 4 \cos \theta }$
D)
$r = \frac { 9 } { 5 + 4 \sin \theta }$
E)
$r = \frac { 9 } { 5 - 4 \cos \theta }$

Question

Quizplus · Accepted Answer

Since the focus is at the origin, we know that the distance from any point on the ellipse to the origin is equal to the distance from that point to the vertex plus the distance from the point to the other vertex. Let the distance from a general point on the ellipse to the vertex at $\left(1,\frac{\pi}{2}ight)$ be $r_1$ and the distance to the other vertex at $(1,-\frac{\pi}{2})$ be $r_2$. Then we have the equation:

$$r_1 + r_2 = 2a$$

where $a$ is the length of the semi-major axis. We also know that the eccentricity $e$ is related to the semi-major axis and the distance between the foci $c$ by the equation $e = \frac{c}{a}$. In this case, $e = 0.8$ and $c = a\cdot 0.8$, so $r_2 - r_1 = 2c = 1.6a$.

Now we can solve for $r_1$ in terms of $r$ and $	heta$ using the formula for converting between Cartesian and polar coordinates: $x = r\cos	heta$ and $y = r\sin	heta$. We have $r_1 = \sqrt{(r\cos	heta-1)^2 + (r\sin	heta-\frac{\pi}{2})^2}$ and $r_2 = \sqrt{(r\cos	heta-1)^2 + (r\sin	heta+\frac{\pi}{2})^2}$, so we can substitute these expressions into the equation above and simplify:

$$\sqrt{(r\cos	heta-1)^2 + (r\sin	heta-\frac{\pi}{2})^2} - \sqrt{(r\cos	heta-1)^2 + (r\sin	heta+\frac{\pi}{2})^2} = \frac{4}{5}r$$

Squaring both sides and simplifying, we get:

$$r^2 - 2r\cos	heta + \frac{1}{4}\left(\pi^2-1-\frac{16}{25}ight) = \frac{16}{25}r^2\sin^2	heta$$

Dividing both sides by $r^2$ and simplifying, we arrive at the polar equation:

$$r = \frac{9}{5+4\sin	heta}$$

which matches choice (D).

Write a Polar Equation In rrr And θ\thetaθ Of an Ellipse with the Focus at the Origin, with Ellipse

Write a Polar Equation In $r$ And $\theta$ Of an Ellipse with the Focus at the Origin, with Ellipse