This document provides information about working with hyperbolas, including: - The standard forms for horizontal and vertical hyperbolas - The Pythagorean property relating the transverse and conjugate axes - Steps for working with hyperbolas from given equations, including writing the equation in standard form, finding axis lengths and coordinates, and sketching the graph - Examples of problems determining equations and properties of hyperbolas, ellipses, and parabolas

1. Working With Hyperbolas
(you have to know where to look)
urban geometry hyperbolas by flickr user johncarney

2. Horizontal Hyperbola Vertical Hyperbola
a
b
b O
a
O

3. The Pythagorean Property
B1 2 2 2
b
c c = a +b
A2 a a A1
F2 O c F1
B2
Horizontal Hyperbola Vertical Hyperbola

4. The Pythagorean Property
2 2 2
c = a +b
a
b
b O
a
O
Horizontal Hyperbola Vertical Hyperbola

5. For the hyperbola whose equation is given below.
(i) Write the equation in standard form
(ii) Determine the lengths of the transverse and conjugate axes,
the coordinates of the verticies and foci, and the equations of the
asymptotes. HOMEWORK
(iii) Sketch a graph of the hyperbola.

6. For the hyperbola whose equation is given below.
(i) Write the equation in standard form
(ii) Determine the lengths of the transverse and conjugate axes,
the coordinates of the verticies and foci, and the equations of the
asymptotes. SLOPE INTERCEPT FORM
(iii) Sketch a graph of the hyperbola.

7. For the hyperbola whose equation is given below.
(i) Write the equation in standard form
(ii) Determine the lengths of the transverse and conjugate axes,
the coordinates of the verticies and foci, and the equations of the
asymptotes. SLOPE INTERCEPT FORM
(iii) Sketch a graph of the hyperbola.

8. The foci of a hyperbola are F (6, 0) and F (-6, 0), and the difference of
1 2
the focal radii is 4 units. Use the definition of a hyperbola to derive the
equation of this hyperbola.

9. A rock is kicked off a vertical cliff and falls in a
parabolic path to the water below. The cliff is 40
m high and the rock hits the water 10 m from the
base of the cliff. What is the horizontal distance
of the rock from the cliff face when the rock is at
a height of 30 m above the water?

10. The cross section of the roof of an indoor tennis court has a semi
elliptical shape. If the roof spans 86 m and has a height of 30 m,
find the height of the roof 20 m from the centre A. HOMEWORK

11. Sketch the graph, identify the vertex, focus, equation of the axis of
symmetry and the equation of the directrix. HOMEWORK

12. A hyperbola has vertices at (1, -4) and (1, 8). If the asymptotes have
slopes ±2, determine the equation of the hyperbola in standard form.
HOMEWORK

13. For each ellipse whose equation is given below
(i) Write the equation in standard form HOMEWORK
(ii) Determine the lengths of the major and minor axes, the
coordinates of the verticies, and the coordinates of the foci.
(iii) Sketch a graph of the ellipse.

14. Determine the equation of a parabola defined by the given conditions.
The vertex is V(-1, 3) and the equation of the directrix is x - 2 = 0
HOMEWORK

15. The cross section of a HOMEWORK
drainage ditch is parabolic in
shape, as shown in the
diagram at the right. When the
width of the water surface is
10 m, the maximum depth of
the water is 1.5 m. Determine
the width of the water, w, when
the maximum depth is 3 m.

16. For each ellipse whose equation is given below
(i) Write the equation in standard form HOMEWORK
(ii) Determine the lengths of the major and minor axes, the
coordinates of the verticies, and the coordinates of the foci.
(iii) Sketch a graph of the ellipse.