# (a) Given the conic section displaystyle{r}=frac{5}{{{7}+{3} cos{{left(thetaright)}}}}, find the x and y intercept(s) and the focus(foci). (b) Given the conic section displaystyle{r}=frac{5}{{{2}+{5} sin{{left(thetaright)}}}}, find the x and y intercept(s) and the focus(foci).

Question
Conic sections
(a) Given the conic section $$\displaystyle{r}=\frac{5}{{{7}+{3} \cos{{\left(\theta\right)}}}}$$, find the x and y intercept(s) and the focus(foci).
(b) Given the conic section $$\displaystyle{r}=\frac{5}{{{2}+{5} \sin{{\left(\theta\right)}}}}$$, find the x and y intercept(s) and the focus(foci).

2020-12-17
(a)
Given $$\displaystyle{r}=\frac{5}{{{7}+{3} \cos{\theta}}}$$
It can be written as $$\displaystyle{r}=\frac{{{5}\text{/}{7}}}{{{1}+{3}\text{/}{7} \cos{\theta}}}$$
Comparing it with general conic section $$\displaystyle{r}=\frac{{{e}{d}}}{{{1}+{e} \cos{\theta}}}$$, we get
$$\displaystyle{e}=\frac{3}{{7}}$$, since e i.e. eccentricity lies between 0 & 1. :. given conic section is ellipse.
for x-intercept $$\displaystyle\theta={0}$$,
$$\displaystyle\pi\ {w}{h}{e}{n}\ \theta={0},{r}=\frac{5}{{10}}=\frac{1}{{2}},$$ so x-intercept in this case is (1/2, 0)
when $$\displaystyle\theta=\pi.{r}=\frac{5}{{4}}$$, so x-intercept in this case is $$\displaystyle{\left(-\frac{5}{{4}},{0}\right)}$$
for y-intercept $$\displaystyle\theta=\frac{\pi}{{2}},\frac{{{3}\pi}}{{2}}$$
when $$\displaystyle\theta=\frac{\pi}{{2}},{r}=\frac{5}{{7}}$$, so y-intercept in this case is $$\displaystyle{\left({0},\frac{5}{{7}}\right)}$$
when $$\displaystyle\theta=\frac{{{3}\pi}}{{2}},{r}=\frac{5}{{7}}$$, so y-intercept in this case is $$\displaystyle{\left({0},-\frac{5}{{7}}\right)}$$
One foci is (0, 0) of this form. Now, to find other foci we will find out center of the ellipse & it will be the mid point of both the x-intercepts
i.e. center $$\displaystyle={\left(\frac{{\frac{1}{{2}}-\frac{5}{{4}}}}{{2}},{0}\right)}$$
$$\displaystyle={\left(-\frac{3}{{8}},{0}\right)}$$
Now we will distance of center $$\displaystyle{\left(-\frac{3}{{8}},{0}\right)}$$ and 1st foci $$(0, 0) , c = 3/8$$
since the distance of both the foci from the center is same, so coordinates of 2nd foci is $$\displaystyle{\left(-\frac{3}{{8}}-\frac{3}{{8}},{0}\right)}$$
i.e. $$\displaystyle{\left(-\frac{6}{{8}},{0}\right)}$$
so, x intercepts are $$\displaystyle{\left(\frac{1}{{2}},{0}\right)},{\left(-\frac{5}{{4}},{0}\right)}$$
y intercepts are $$\displaystyle{\left({0},\frac{5}{{7}}\right)},{\left({0},-\frac{5}{{7}}\right)}$$
foci are $$\displaystyle{\left({0},{0}\right)},{\left(-\frac{6}{{8}},{0}\right)}$$
(b)
Given $$\displaystyle{r}=\frac{5}{{{2}+{5} \sin{\theta}}}$$
It can be written as $$\displaystyle{r}=\frac{{{5}\text{/}{2}}}{{{1}+{5}\text{/}{2} \sin{\theta}}}$$
Comparing it with general conic section $$\displaystyle{r}=\frac{{{e}{d}}}{{{1}+{e} \sin{\theta}}}$$, we get
$$\displaystyle{e}=\frac{5}{{2}}$$, since e i.e. eccentricity is greater than $$\displaystyle{1}.\therefore$$ given conic section is hyperbola.
for x-intercept $$\displaystyle\theta={0},\pi$$
when $$\displaystyle\theta={0},{r}=\frac{5}{{2}}$$, so x -intercept in this case is $$\displaystyle{\left(\frac{5}{{2}},{0}\right)}$$
when $$\displaystyle\theta=\pi.{r}=\frac{5}{{2}}$$,so x -intercept in this case is $$\displaystyle{\left(-\frac{5}{{2}},{0}\right)}$$
for y-intercept $$\displaystyle\theta=\frac{\pi}{{2}},\frac{{{3}\pi}}{{2}}$$
when $$\displaystyle\theta=\frac{\pi}{{2}},{r}=\frac{5}{{7}}$$, so y-intercept in this case is $$\displaystyle{\left({0},\frac{5}{{7}}\right)}$$
when $$\displaystyle\theta=\frac{{{3}\pi}}{{2}},{r}=-\frac{5}{{3}}$$, since r is negative .so y-intercept in this case is $$\displaystyle{\left({0},\frac{5}{{3}}\right)}$$
One foci is (0, 0) of this form. Now, to find other foci we will find out center of the hyperbola & it will be the mid point of both the y-intercepts
i.e. center $$\displaystyle={\left({0},\frac{{\frac{5}{{7}}+\frac{5}{{3}}}}{{2}}\right)}$$
$$\displaystyle={\left({0},\frac{25}{{21}}\right)}$$
Now we will distance of center $$\displaystyle{\left({0},\frac{25}{{21}}\right)}$$ and 1st foci $$\displaystyle{\left({0},{0}\right)},{c}=\frac{25}{{21}}$$
since the distance of both the foci from the center is same, so coordinates of 2nd foci is $$\displaystyle{\left({0},\frac{25}{{21}}+\frac{25}{{21}}\right)}{i}.{e}.{\left({0},\frac{50}{{21}}\right)}$$
so, x intercepts are $$\displaystyle{\left(\frac{5}{{2}},{0}\right)},{\left(-\frac{5}{{2}},{0}\right)}$$
y intercepts are $$\displaystyle{\left({0},\frac{5}{{7}}\right)},{\left({0},\frac{5}{{7}}\right)}$$
foci are $$\displaystyle{\left({0},{0}\right)},{\left({0},\frac{50}{{21}}\right)}$$

### Relevant Questions

Identify the conic section given by $$\displaystyle{y}^{2}+{2}{y}={4}{x}^{2}+{3}$$
Find its $$\frac{\text{vertex}}{\text{vertices}}\ \text{and}\ \frac{\text{focus}}{\text{foci}}$$
Solve, a.Determine the conic section of the polar equation $$\displaystyle{r}=\frac{8}{{{2}+{2} \sin{\theta}}}$$ represents. b. Describe the location of a directrix from the focus located at the pole.
A latus rectum of a conic section is a chord through a focus parallel to the directrix. Find the area bounded by the parabola $$\displaystyle{y}={x}^{2}\text{/}{\left({4}{c}\right)}$$ and its latus rectum.
Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
b) Write the equation of a hyperbola with vertices at (3, 3) and (-3,3). The foci are located at (4, 3) and (-4, 3).
c) Write the equation of a parabola with vertex at (-2,4) and focus at (-4, 4)
Write the equation of each conic section, given the following characteristics:
a) Write the equation of an ellipse with center at (3, 2) and horizontal major axis with length 8. The minor axis is 6 units long.
b) Write the equation of a hyperbola with vertices at (3, 3) and (-3, 3). The foci are located at (4, 3) and (-4, 3).
c) Write the equation of a parabola with vertex at (-2, 4) and focus at (-4, 4)
To find the vertices and foci of the conic section: $$\displaystyle{\frac{{{\left({x}\ -\ {4}\right)}^{{{2}}}}}{{{5}^{{{2}}}}}}\ -\ {\frac{{{\left({y}\ +\ {3}\right)}^{{{2}}}}}{{{6}^{{{2}}}}}}={1}$$
Eliminate the parameter and obtain the standard form of the rectangular equation. Circle: $$x = h + r cos(?), y = k + r sin(?)$$ Use your result to find a set of parametric equations for the line or conic section. $$(When 0 \leq ? \leq 2?.)$$ Circle: center: (6, 3), radius: 7
Identify the conic section given by the polar equation $$\displaystyle{r}=\frac{4}{{{1}- \cos{\theta}}}$$ and also determine its directrix.
$$\displaystyle{\left({a}\right)}{4}{x}{2}-{9}{y}{2}={12}{\left({b}\right)}-{4}{x}+{9}{y}{2}={0}$$
$$\displaystyle{\left({c}\right)}{4}{y}{2}+{9}{x}{2}={12}{\left({d}\right)}{4}{x}{3}+{9}{y}{3}={12}$$
Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work. $$\displaystyle{r}=\ {\frac{{{10}}}{{{5}\ +\ {2}\ {\cos{\theta}}}}}$$