Hyperbolas Conic Number Three (10.3)

POD– Relationships What is the relationship between a, b, and c in an ellipse? There is another special relationship between a and c for all conics. The ratio between the length from center to focus, and the length from center to vertex is called the eccentricity (e) of the ellipse. In other words, e = c/a. Eccentricity ranges from 0 to 1. Which of these would be with a flatter ellipse, and which a rounder ellipse?

Another way to view conics Look at the handout. On the front are various diagrams of planes, cones and the conic intersections. On the back is a way to view conics using a focus and a directrix. Notice how eccentricity is defined here. We’ll use that to look at Unified Conics in GSP.

Another way to view conics

Focus and directrix When we look at hyperbolas and ellipses, those distances are different. If d 1 = distance from focus to curve, and d 2 = distance from directrix to curve, In an ellipse, d 1 < d 2, and the curve comes back in on itself. In a hyperbola, d 1 > d 2, and the curve expands out without bound.

Focus and directrix We’ve seen eccentricity (e) as the ratio c/a in an ellipse. It can also be described as the ratio d 1 /d 2. In a parabola, d 1 = d 2, so e = 1. In an ellipse, d 1 < d 2, so e < 1. In a hyperbola, d 1 > d 2, so e > 1. What is the difference between these two ways of looking at eccentricity?

Focus and directrix We’ve seen eccentricity (e) as the ratio c/a in an ellipse. It can also be described as the ratio d 1 /d 2. In a parabola, d 1 = d 2, so e = 1. So c/a = 1. In an ellipse, d 1 < d 2, so e < 1. So c/a < 1. In a hyperbola, d 1 > d 2, so e > 1. So c/a > 1. What is the difference between these two ways of looking at eccentricity?

Review—Ellipses What do the variables represent? What is the necessary condition for the coefficients below?

Hyperbolas What changes from the ellipse equations for hyperbolas? What do the variables represent? What is the necessary condition for the coefficients?

Hyperbolas Could we have either form of this equation? How are they different?

Ellipses and hyperbolas Sketch a general shape for the ellipses on the origin. Sketch a general shape for hyperbolas. Where are the foci located in each of these conics? Notice how the order changes: Ellipse: VFCFV Hyperbola: FVCVF

Ellipses Based on our hands on activity with styrofoam, we have this definition for an ellipse: The set of all points on a plane with a constant sum of distances from two given points (the foci). What sort of change would we make for a hyperbola?

Ellipses and hyperbolas Based on our hands on activity with styrofoam, we have this definition for an ellipse: The set of all points on a plane with a constant sum of distances from two given points (the foci). What sort of change would we make for a hyperbola? The set of all points on a plane with a constant difference of distances between two given points (the foci).

Ellipses and hyperbolas The radii in an ellipse are called major and minor. The radii for a hyperbola are called transverse and conjugate. Based on our earlier sketch of a hyperbola and its foci, what do you think changes in the relationship of radii with hyperbolas?

Ellipses and hyperbolas In ellipses: c 2 = a 2 – b 2 In hyperbolas: c 2 = a 2 +b 2 Which makes sense, since now the focal radius is the longest radius.

Graphs of hyperbolas In order to identify the parts of a hyperbola, we’ll start by graphing one. The algebra to move from one form of equation to the other is the same as that we used for ellipses. The graphing changes.

Graphs of hyperbolas Start simple. Give the center, major and minor radii, and calculate the focal radius. Which way would this open?

Graphs of hyperbolas Start simple. Major radius (a) = 3 Minor radius (b) = 2 Focal radius (c)= √13 Opens sideways.

Graphs of hyperbolas Identify the parts: Transverse axis Conjugate axis (How do we determine which is which?) Foci Vertices (Which axis do these ride on?)

Graphs of hyperbolas What would this hyperbola look like in comparison?

Graphs of hyperbolas What would this hyperbola look like in comparison? Same shape, center moved to (-5,2).

Try the reverse Find an equation for the graph of a hyperbola with vertices at (±3, 0), and that contains the point (5,2).

Try the reverse Find an equation for the graph of a hyperbola with vertices at (±3, 0) and that contains the point (5,2). First, where is the center, and how do you know? What it its orientation?

Try the reverse Find an equation for the graph of a hyperbola with vertices at (±3, 0), and that contains the point (5,2). First, where is the center, and how do you know? At the origin, because the vertices are centered on the point (0,0). It’s oriented sideways. Now, the numbers.

Try the reverse Find an equation for the graph of a hyperbola with vertices at (±3, 0), and that contains the point (5,2). Center (0, 0) Now, the numbers. What radius do we know? How do we find the other?

Try the reverse Find an equation for the graph of a hyperbola with vertices at (±3, 0), and that contains the point (5,2). Center (0, 0) Now, the numbers. What radius do we know? The transverse axis is sideways, and the distance from vertex to vertex is 6, so the x-radius is 3. How do we find the other? Create as much of an equation as possible and plug the point (5, 2) into it.

Try the reverse Find an equation for the graph of a hyperbola with vertices at (±3, 0), and that contains the point (5,2). Center (0, 0) x-radius = 3 How do we find the other? Create as much of an equation as possible and plug the point (5, 2) into it.

Try the reverse Find an equation for the graph of a hyperbola with vertices at (±3, 0), and that contains the point (5,2). Center (0, 0) x-radius = 3 y-radius = 3/2

Swinging form one form to another Graph this equation. How do you know it’s a hyperbola?

Swinging form one form to another Graph this equation. Complete the square.

Swinging form one form to another Graph this equation. Now graph it.

Swinging form one form to another Graph this equation. Now graph it.

Asymptotes What are the equations for the asymptotes?

Asymptotes What are the equations for the asymptotes? y-6 = ±2(x+2) Give a general statement about the slopes of the asymptotes.

Asymptotes What are the equations for the asymptotes? y-6 = ±2(x+2) Give a general statement about the slopes of the asymptotes.