Pre-Cal 40S Slides December 17, 2007

12.3 Hyperbolas

The Anatomy of the Hyperbola

The Standard Form for the Equation of a Hyperbola

Horizontal Orientation Vertical Orientation

Similarities Differences

• all variables are squared • they both equal 1• denominators tell you the semi-conjugate and semi-transverse axes• both equations are differences• "h" is always with "x", and "k" is always with "y"• they both tell you the centre (h,k)

• the positive term in a horizontal hyperbola is the x term for a vertical hyperbola y is positive• the denominator has switched• a is underneath y in a vertical hyperbola, a is underneath x in the horizontal hyperbola

Conics Animations Source

Conics Animations Source

Conics Animations Source

Conics Animations Source

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.(iii) Sketch a graph of the hyperbola.

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.(iii) Sketch a graph of the hyperbola. SLOPE INTERCEPT FORM

For each ellipse whose equation is given below

(i) Write the equation in standard form(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.

The foci of an ellipse are F1(2, 3) and F2(-2, 3), and the sum of the focal radii is 6 units. Use the definition of an ellipse to derive the equation of this ellipse.

Find the radius, and the coordinates of the centre of the circle:

The foci of a hyperbola are F1(6, 0) and F2(-6, 0), and the difference of the focal radii is 4 units. Use the definition of a hyperbola to derive the equation of this hyperbola.

Determine the equaiton of a parabola defined by the given conditions.

The vertex is V(-1, 3) and the equation of the directrix is x - 2 = 0

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?

A hyperbola has centre (0, 0) and one vertex A .

(c) Find the value of b if L(3, b) is on the hyperbola.

(b) Find the value of a if K(a, 2) is on the hyperbola.

(a) Find the equation of the hyperbola if it passes through J(9, 5)

Consider the parabola y2 - 20x + 2y + 1 = 0. Write the equation in standard form, find the coordinates of the vertex and focus and the equation of the directrix.

Write the equation of the circle x2 + y2 + 2x - 10y + 25 = 0 in standard form and find the centre and the radius.

The parabola y2 - x + 4y + k = 0 passes through the point (12, 1). Find the coordinates of the vertex and sketch the graph.

A point P(x, y) moves such that it is always equidistant from the point A(2, 3) and the line y = -1. Determine the equation of this locus in standard form.

An ellipse has centre (-2, 4) and one vertex A(8, 4).

(a) Find the equation of the ellipse if it passes through R(4, 8).(b) Find the value of c if S(c, 7) is on the ellipse.(c) Find the value of d if T(3, d) is on the ellipse.