# 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}}

Question
Conic sections
Identify the conic section given by $$\displaystyle{y}^{2}+{2}{y}={4}{x}^{2}+{3}$$
Find its $$\frac{\text{vertex}}{\text{vertices}}\ \text{and}\ \frac{\text{focus}}{\text{foci}}$$

2021-02-11
Step 1
We rewrite the equation as:
$$\displaystyle{y}^{2}+{2}{y}={4}{x}^{2}+{3}$$
$$\displaystyle{y}^{2}+{2}{y}-{4}{x}^{2}={3}$$
$$\displaystyle{y}^{2}+{2}{y}+{1}-{4}{x}^{2}={3}+{1}$$
$$\displaystyle{\left({y}+{1}\right)}^{2}-{4}{x}^{2}={4}$$
$$\displaystyle\frac{{{\left({y}+{1}\right)}^{2}}}{{4}}-\frac{{{4}{x}^{2}}}{{4}}={1}$$
$$\displaystyle\frac{{{\left({y}+{1}\right)}^{2}}}{{4}}-\frac{{x}^{2}}{{1}}={1}$$
This is an equation of a hyperbola.
Step 2
Then we compare the equation with standard form.
$$\displaystyle\frac{{{\left({y}+{1}\right)}^{2}}}{{4}}-\frac{{x}^{2}}{{1}}={1}$$
$$\displaystyle\frac{{{\left({y}-{k}\right)}^{2}}}{{b}^{2}}-\frac{{{\left({x}-{h}\right)}^{2}}}{{a}^{2}}={1}$$
$$\displaystyle{h}={0},{k}=-{1},$$
$$\displaystyle{a}^{2}={1},{b}^{2}={4}$$
$$\displaystyle{a}-{1},{b}={2}$$
$$\text{vertex}\ \displaystyle={\left({h},{k}\pm{b}\right)}={\left({0},-{1}\pm{2}\right)}={\left({0},{1}\right)},{\left({0},-{3}\right)}$$
$$\text{Foci}\ \displaystyle={\left({h},{k}\pm\sqrt{{{a}^{2}+{b}^{2}}}={\left({0},-{1}\pm\sqrt{{{1}+{4}}}={\left({0},-{1}+\sqrt{{5}}\right)},{\left({0},-{1}-\sqrt{{5}}\right)}\right.}\right.}$$
$$\text{vertex}\ \displaystyle={\left({0},{1}\right)},{\left({0},{3}\right)}$$
$$\text{Foci}\ \displaystyle={\left({0},-{1}+\sqrt{{5}}\right)},{\left({0},-{1}-\sqrt{{5}}\right)}$$

### Relevant Questions

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).
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.
To find the vertices and foci of the conic section: $$\displaystyle{\frac{{{\left({x}\ -\ {4}\right)}^{{{2}}}}}{{{5}^{{{2}}}}}}\ -\ {\frac{{{\left({y}\ +\ {3}\right)}^{{{2}}}}}{{{6}^{{{2}}}}}}={1}$$
Identify the conic section given by the polar equation $$\displaystyle{r}=\frac{4}{{{1}- \cos{\theta}}}$$ and also determine its directrix.
Find the center, foci, vertices, asymptotes, and radius, if necessary, of the conic sections in the equation:
$$\displaystyle{x}^{2}+{4}{x}+{y}^{2}={12}$$
Convert the following equations to its standard form and identify the vertex and focus.
$$\displaystyle{3}{y}^{2}+{8}{x}+{24}{y}+{40}={0}$$
To determine: The conic section and to find the vertices and foci: $$\displaystyle{x}{2}\ -\ {y}{2}{y}={4}$$
a) determine the type of conic b) find the standard form of the equation Parabolas: vertex, focus, directrix Circles: Center, radius Ellipses: center, vertices, co-vertices, foci Hyperbolas: center, vertices, co-vertices, foci, asymptotes $$16x^2 + 64x - 9y^2 + 18y - 89 = 0$$
Find the equation of the graph for each conic in standard form. Identify the conic, the center, the co-vertex, the focus (foci), major axis, minor axis, $$a^{2}, b^{2},\ and c^{2}.$$ For hyperbola, find the asymptotes $$9x^{2}\ -\ 4y^{2}\ +\ 54x\ +\ 32y\ +\ 119 = 0$$