# Determine the equation of a conic section...(Hyperbola) Given: center (-9, 1) distance between displaystyle{F}{1}{quadtext{and}quad}{F}{2}={20} units distance between displaystyle{C}{V}{1}{quadtext{and}quad}{C}{V}{2}={4} units orientation = vertical

Question
Conic sections
Determine the equation of a conic section...(Hyperbola) Given: center (-9, 1) distance between $$\displaystyle{F}{1}{\quad\text{and}\quad}{F}{2}={20}$$ units
distance between $$\displaystyle{C}{V}{1}{\quad\text{and}\quad}{C}{V}{2}={4}$$ units orientation = vertical

2021-02-06
Step 1
Given: center (-9, 1)
distance between $$\displaystyle{F}{1}{\quad\text{and}\quad}{F}{2}={20}$$ units
distance between $$\displaystyle{C}{V}{1}{\quad\text{and}\quad}{C}{V}{2}={4}$$ units
orientation = vertical
Step 2
Standard equation of hyperbola
$$\displaystyle\frac{{{\left({y}-{k}\right)}^{2}}}{{a}^{2}}-\frac{{{\left({x}-{h}\right)}^{2}}}{{b}^{2}}={1}$$
where (h,k) is centre and orientation is vertical
We know distance between $$\displaystyle{F}{1}{\quad\text{and}\quad}{F}{2}={2}{C}={20}$$
so $$\displaystyle{C}={10}\ \text{unit and}\ {c}^{2}={a}^{2}+{b}^{2}$$
here center is (-9, 1)
$$\displaystyle{h}=-{9}{\quad\text{and}\quad}{k}={1}$$
length of $$\displaystyle{C}{V}{1}\to{C}{V}{2}={2}{b}$$
$$\displaystyle{2}{b}={4}$$ unit
$$\displaystyle{b}={2}$$ unit
Now from $$\displaystyle{c}^{2}={a}^{2}+{b}^{2}$$
$$\displaystyle{\left({10}\right)}^{2}={a}^{2}+{\left({2}\right)}^{2}$$
$$\displaystyle{100}-{4}={a}^{2}$$
$$\displaystyle{a}=\sqrt{{96}}$$
Step 3
So the equation of hyperbola is
$$\displaystyle\frac{{\left({y}-{1}\right)}^{2}}{{96}}-\frac{{\left({x}+{9}\right)}^{2}}{{4}}={1}$$
where (-9, 1) is centre

### 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)
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)
Decide if the equation defines an ellipse, a hyperbola, a parabola, or no conic section at all.
$$\displaystyle{\left({a}\right)}{4}{x}{2}-{9}{y}{2}={12}{\left({b}\right)}-{4}{x}+{9}{y}{2}={0}$$
$$\displaystyle{\left({c}\right)}{4}{y}{2}+{9}{x}{2}={12}{\left({d}\right)}{4}{x}{3}+{9}{y}{3}={12}$$
Solve for the equation in standard form of the following conic sections and graph the curve on a Cartesian plane indicating important points.
1. An ellipse passing through (4, 3) and (6, 2)
2. A parabola with axis parallel to the x-axis and passing through (5, 4), (11, -2) and (21, -4)
3. The hyperbola given by $$\displaystyle{5}{x}^{2}-{4}{y}^{2}={20}{x}+{24}{y}+{36}.$$
For Exercise, an equation of a degenerate conic section is given. Complete the square and describe the graph of each equation.
$$\displaystyle{9}{x}{2}+{4}{y}{2}-{24}{y}+{36}={0}$$
Identify the conic section given by the polar equation $$\displaystyle{r}=\frac{4}{{{1}- \cos{\theta}}}$$ and also determine its directrix.
$$\displaystyle{2}{x}{2}-{8}{x}{y}+{3}{y}{2}-{4}={0}$$
$$\displaystyle{2}{x}^{2}={y}^{2}+{2}$$
$$\displaystyle{x}^{2}-{2}{x}{y}+{y}^{2}+{x}-{2}{y}-{1}={0}$$
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.