12.2 The Hyperbola

Use the distance formula to find the distance between the points (−5, −3) and (7,2).

Identify the vertices and foci of the hyperbola with equation $\frac{y^{2}}{49} - \frac{x^{2}}{32} = 1.$

Identify the vertices and foci of the hyperbola with equation $\frac{x^{2}}{9} - \frac{y^{2}}{25} = 1.$

What is the standard form equation of the hyperbola that has vertices $\left(0, \pm 2\right)$ and foci $\left(0, \pm 2 \sqrt{5}\right) ?$

What is the standard form equation of the hyperbola that has vertices at $\left(0 , −2\right)$ and $\left(6 , −2\right)$ and foci at $\left(−2 , −2\right)$ and $\left(8 , −2\right) ?$

What is the standard form equation of the hyperbola that has vertices $\left(1 , −2\right)$ and $\left(1 , \text{8}\right)$ and foci $\left(1 , −10\right)$ and $\left(1 , 16\right) ?$

Define a hyperbola in terms of its foci.

What must be true of the foci of a hyperbola?

Where must the center of hyperbola be relative to its foci?

Determine whether the following equations represent hyperbolas. If so, write in standard form.

$\frac{x^{2}}{36} - \frac{y^{2}}{9} = 1$

Determine whether the following equations represent hyperbolas. If so, write in standard form.

$25 x^{2} - 16 y^{2} = 400$

Write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.

$\frac{x^{2}}{25} - \frac{y^{2}}{36} = 1$

Write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.

$\frac{y^{2}}{4} - \frac{x^{2}}{81} = 1$

Write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.

$\frac{\left(x - 1\right)^{2}}{9} - \frac{\left(y - 2\right)^{2}}{16} = 1$

Write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.

$\frac{\left(x - 2\right)^{2}}{49} - \frac{\left(y + 7\right)^{2}}{49} = 1$

Write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.

$- 9 x^{2} - 54 x + 9 y^{2} - 54 y + 81 = 0$

Write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.

$- 4 x^{2} + 24 x + 16 y^{2} - 128 y + 156 = 0$

Write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.

$x^{2} + 2 x - 100 y^{2} - 1000 y + 2401 = 0$

Write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.

$4 x^{2} + 24 x - 25 y^{2} + 200 y - 464 = 0$

Find the equations of the asymptotes for each hyperbola.

$\frac{\left(x - 3\right)^{2}}{5^{2}} - \frac{\left(y + 4\right)^{2}}{2^{2}} = 1$

Find the equations of the asymptotes for each hyperbola.

$9 x^{2} - 18 x - 16 y^{2} + 32 y - 151 = 0$

Given information about the graph of the hyperbola, find its equation.

Vertices at $\left(3 , 0\right)$ and $\left(−3 , 0\right)$ and one focus at $\left(5 , 0\right) .$

Given information about the graph of the hyperbola, find its equation.

Vertices at $\left(1 , 1\right)$ and $\left(11 , 1\right)$ and one focus at $\left(12 , 1\right) .$

Given information about the graph of the hyperbola, find its equation.

Center: $\left(4 , 2\right) ;$ vertex: $\left(9 , 2\right) ;$ one focus: $\left(4 + \sqrt{26} , 2\right) .$

Assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.

The object enters along a path approximated by the line $y = 2 x - 2$ and passes within 0.5 au of the sun at its closest approach, so the sun is one…

Assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.

The object enters along a path approximated by the line $y = \frac{1}{3} x - 1$ and passes within 1 au of the sun at its closest approach, so the sun is one fo…