# To find the vertices and foci of the conic section: frac{(x - 4)^{2}}{5^{2}} - frac{(y + 3)^{2}}{6^{2}}=1

Question
Conic sections
To find the vertices and foci of the conic section: $$\displaystyle{\frac{{{\left({x}\ -\ {4}\right)}^{{{2}}}}}{{{5}^{{{2}}}}}}\ -\ {\frac{{{\left({y}\ +\ {3}\right)}^{{{2}}}}}{{{6}^{{{2}}}}}}={1}$$

2021-02-21
Concept used: Write the general equation of the hyperbola. $$\displaystyle{\frac{{{\left({x}-{h}\right)}^{{{2}}}}}{{{a}^{{{2}}}}}}-{\frac{{{\left({y}-{k}\right)}^{{{2}}}}}{{{b}^{{{2}}}}}}={1}$$ The centre of the hyperbola is (h,k), the vertices are $$\displaystyle{\left({h}\pm{a},{k}\right)}$$, and foci are $$\displaystyle{\left({h}\pm{c},{k}\right)}$$ Write the expression for c. $$\displaystyle{c}=\sqrt{{{a}^{{{2}}}+{b}^{{{2}}}}}$$ The given equation is, $$\displaystyle{\left({\frac{{{x}-{4}}}{{{5}}}}\right)}^{{{2}}}-{\left({\frac{{{y}+{3}}}{{{6}}}}\right)}^{{{2}}}={1}$$
$$\displaystyle{\frac{{{\left({x}-{4}\right)}^{{{2}}}}}{{{5}^{{{2}}}}}}-{\frac{{{\left({y}+{3}\right)}^{{{2}}}}}{{{6}^{{{2}}}}}}={1}$$ Write the general equation of the hyperbola. $$\displaystyle{\frac{{{\left({x}-{h}\right)}^{{{2}}}}}{{{a}^{{{2}}}}}}-{\frac{{{\left({y}-{k}\right)}^{{{2}}}}}{{{b}}}}={1}$$ Compare the given equation with the general equation of the hyperbola to get, $$\displaystyle{h}={4}$$
$$\displaystyle{k}=-{3}$$
$$\displaystyle{a}={5}$$
$$\displaystyle{b}={6}$$ Write the expression for c. $$\displaystyle{c}=\sqrt{{{a}^{{{2}}}+{b}^{{{2}}}}}$$ Substitute 6 for a and 5 for b. $$\displaystyle{c}=\sqrt{{{5}^{{{2}}}+{6}^{{{2}}}}}$$
$$\displaystyle=\sqrt{{{25}+{36}}}$$
$$\displaystyle=\sqrt{{{61}}}$$ The vertices of the conic section are, $$\displaystyle{\left({h}\pm{a},{k}\right)}={\left({4}\pm{5},-{3}\right)}$$
$$\displaystyle={\left({4}+{5},-{3}\right)}{\left({4}-{5},-{3}\right)}$$
$$\displaystyle={\left({9},-{3}\right)}{\left(-{1},-{3}\right)}$$ The foci of the conic section are, $$\displaystyle{\left({h}\pm{a},{k}\right)}={\left({4}\pm\sqrt{{{61}}},-{3}\right)}$$
$$\displaystyle={\left({4}+\sqrt{{{61}}},-{3}\right)}{\left({4}-\sqrt{{{61}}},-{3}\right)}$$ Therefore, the vertices and foci of the conic section are $$\displaystyle{\left({9},-{3}\right)}{\left(-{1},-{3}\right)}$$ and PSK(4+\sqrt{61},-3)(4-\sqrt{61},-3) respectively.

### Relevant Questions

Identify the conic section given by $$\displaystyle{y}^{2}+{2}{y}={4}{x}^{2}+{3}$$
Find its $$\frac{\text{vertex}}{\text{vertices}}\ \text{and}\ \frac{\text{focus}}{\text{foci}}$$
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)
(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).
To determine: The conic section and to find the vertices and foci: $$\displaystyle{x}{2}\ -\ {y}{2}{y}={4}$$
Find the center, foci, vertices, asymptotes, and radius, if necessary, of the conic sections in the equation:
$$\displaystyle{x}^{2}+{4}{x}+{y}^{2}={12}$$
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. $$\displaystyle{r}=\ {\frac{{{10}}}{{{5}\ +\ {2}\ {\cos{\theta}}}}}$$
To calculate: The vertices and foci of the conic section: $$\displaystyle{x}{29}\ +\ {y}{24}={1}$$
The current in a wire varies with time according to the relation $$I=55A?(0.65A/s2)t2$$.
How many coulombs of charge pass a cross section of the wire in the time interval between $$t=0$$ and $$t = 8.5s$$ ?
3. The hyperbola given by $$\displaystyle{5}{x}^{2}-{4}{y}^{2}={20}{x}+{24}{y}+{36}.$$
To calculate the verties and foci of the conic section: $$\displaystyle{\left({x}{9}\right)}{2}\ +\ {\left({y}{4}\right)}{2}={1}$$