(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
asked 2020-12-16
(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).

Answers (1)

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 \(\displaystyle{\left({0},{0}\right)},{\left({0},\frac{50}{{21}}\right)}\)
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