# To calculate the verties and foci of the conic section: (x9)2 + (y4)2=1

Question
Conic sections
To calculate the verties and foci of the conic section: $$\displaystyle{\left({x}{9}\right)}{2}\ +\ {\left({y}{4}\right)}{2}={1}$$

2021-03-08
Step 1 Formula: eccentricity $$\displaystyle{\left({e}\right)}={1}\ -\ {b}{2}{a}{2}$$ Step 2 Calculation: Compare this equation with the standard ellipse equation $$\displaystyle{x}{2}{a}{2}\ +\ {y}{2}{b}{2}={1}$$ and we get: $$\displaystyle{a}{2}={81}\ \Rightarrow\ {a}={9}{b}{2}={16}\ \Rightarrow\ {b}={4}$$ Now, $$\displaystyle{e}={1}\ -\ {b}{2}{a}{2}={1}\ -\ {1681}={659}$$ Vertices are: $$\displaystyle{\left(\pm\ {a},\ {0}\right)}\ \text{and}\ {\left({0},\ \pm\ {b}\right)}$$ Now, put the values of a and b to get the vertices of the conic section and we get: $$\displaystyle{\left(\pm\ {a},\ {0}\right)}\ \text{and}\ {\left({0},\ \pm\ {b}\right)}={\left(\pm\ {9},\ {0}\right)}\ \text{and}\ {\left({0},\ \pm\ {4}\right)}$$ Now, $$\displaystyle\text{Foci}\ ={\left(\pm\ {a}{e},\ {0}\right)}={\left(\pm\ {65},\ {0}\right)}$$ Thus, the vertices and foci of the conic section are: $$\displaystyle{\left(\pm\ {9},\ {0}\right)}\ \text{and}\ {\left({0},\ \pm\ {4}\right)}\ \text{and}\ {\left(\pm\ {65},\ {0}\right)}.$$

### Relevant Questions

To calculate: The vertices and foci of the conic section: $$\displaystyle{x}{29}\ +\ {y}{24}={1}$$
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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)
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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)
(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).
Identify the conic section given by $$\displaystyle{y}^{2}+{2}{y}={4}{x}^{2}+{3}$$
Find its $$\frac{\text{vertex}}{\text{vertices}}\ \text{and}\ \frac{\text{focus}}{\text{foci}}$$
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$$\displaystyle{{x}_{{1}}^{{2}}}-{12}{x}_{{1}}{x}_{{2}}+{{x}_{{2}}^{{2}}}={70}$$
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