9.3 Hyperbolas

Given two points F₁ and F₂ in the plane lying a distance 2c apart and given a distance 2a < 2c, the set of points whose distances to F₁ and F₂ respectively differ by 2a is a hyperbola. F₁ and F₂ are the foci of the hyperbola.

Exercise 9.3.1

Get a compass and a blank sheet of paper. On the sheet of paper, mark two points F₁ and F₂ and draw a dotted line through them. Construct a dotted perpendicular line to F₁F₂ through the midpoint of that segment. Label the midpoint as the center. Place the point of the compass at the center and open the compass to a fixed radius less than the distance from the center to the foci. Mark points V₁ and V₂ where the pencil of the compass crosses the line that passes through F₁ and F₂. These are the vertices of the hyperbola. Construct vertical dotted lines through V₁ and V₂ . Place the point of the compass at the center and open it to a radius equal to the distance from the center to either focus, a distance c. Mark the points on the verticals through V₁ and V₂ where the pencil of the compass crosses. Connect those four points into a dotted rectangle. Using dotted lines, extend the diagonals of that rectangle. These are the two asymptotes of the hyperbola. From each of the vertices, draw a line curving away from the center and toward each of the asymptotes. The hyperbola is a graph which is in two separate pieces. The line through the foci is the major axis and the line through the center perpendicular to the major axis is the transverse axis.

Solution

Exercise 9.3.2

Using the hyperbola that you drew in Exercise 9.3.1, denote the distance from the center to V₁ and V₂ as a. Find the difference of the distances from V₁ to F₁ and V₁ to F₂ in terms of a. Do the same for the sum of the distances from V₂ to F₁ and V₂ to F₂.

Solution

Exercise 9.3.3

The width of the central rectangle you drew in Exercise 9.3.1 is 2a, and the length of its diagonals is 2c. If its height is 2b, find the relationship between a, b and c.

Solution

We will consider only hyperbola with either a vertical or horizontal major axis.

Let us begin by considering a hyperbola with center at the origin and foci ( ± c, 0 ) on the x-axis. Then the x axis is the major axis. Let ( ± a, 0 ) denote the vertices and ( 0, ± b ) the points where the central rectangle cross the y-axis. Construct the central rectangle. The distance from the center ( 0, 0 ) to the corners of the central rectangle is the same as the distance to the foci ( ± c, 0 ). The equation of this ‘horizontal’ hyperbola is

If this hyperbola is shifted h units horizontally and k units vertically, the resulting ellipse will have equation

Next, let us consider a hyperbola with center at the origin and foci ( 0, ± c ) on the y-axis. Then the y axis is the major axis. Let ( 0, ± a ) denote the vertices and ( ± b, 0 ) the points where the central rectangle cross the x-axis. Construct the central rectangle. The distance from the center ( 0, 0 ) to the corners of the central rectangle is the same as the distance to the foci. The equation of this ‘vertical’ hyperbola is

If this hyperbola is shifted h units horizontally and k units vertically, the resulting ellipse will have equation

In either case c² = a² + b².

Exercise 9.3.4

Find the equation of the hyperbola with center ( 0, 0 ), vertex ( -3, 0 ) and focus ( 5, 0 ).

Solution

Exercise 9.3.5

Find the equation of the hyperbola with center ( 3, -2 ), vertex ( 3, -3 ) and focus ( 3, 0 ).

Solution

Be able to complete the square to put the equation of a hyperbola in standard form. Be able to find the center, foci and vertices of a hyperbola, given its equation.

Exercise 9.3.6

Complete the square to put the equation into standard form, then find the center, vertices and foci.

4x² - y² + 16x + 2y + 19 = 0

Solution

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