Hyperbolas Page 6

Page 6

Using the Standard Equation of a Hyperbola Centred at (h, k)
to Describe its Asymptotes and Sketch its Graph

You know that the values of a and b affect the slopes of the asymptotes in the equation of a hyperbola not centred at the origin.
How can we use the standard equations (x - h)²/ a² - (y - k)²/ b² = 1 and -(x - h)² / a² + (y - k)²/ b² = 1 to describe the asymptotes?
The method is the same as when we worked with the standard equations x²/ a² - y²/ b² = 1 and -x²/ a² + y²/ b² = 1.
Again we see that when the vertices of the hyperbola are on the horizontal axis:
the distance between the vertices is 2a.
the distance between the centre of the hyperbola and each vertex is |a|.
there exists points on the vertical axis such that the distance between each point and the centre of the hyperbola is |b|.
When the vertices of the hyperbola are on the vertical axis:
the distance between the vertices is 2b.
the distance between the centre of the hyperbola and each vertex is |b|.
there exists points on the horizontal axis is such that the distance between each point and the centre is |a|.
As before, we can draw a rectangle of width 2a and height 2b in the graph, and use this rectangle to find that the slopes of the asymptotes are +b/a and -b/a.

Once you know the slopes of the asymptotes and centre of the hyperbola, sketch them on your graph paper. Then determine the locations of the vertices and go ahead and sketch the hyperbola that passes through the vertices and approaches the asymptotes. This method is more closely examined in the questions below.

(x - h)² - (y - k)² = 1 or - (x - h)² + (y - k)² = 1

a² b² a² b²

You know that the values of a and b affect the slopes of the asymptotes in the equation of a hyperbola not centred at the origin.
How can we use the standard equations (x - h)²/ a² - (y - k)²/ b² = 1 and -(x - h)² / a² + (y - k)²/ b² = 1 to describe the asymptotes?
The method is the same as when we worked with the standard equations x²/ a² - y²/ b² = 1 and -x²/ a² + y²/ b² = 1.
Again we see that when the vertices of the hyperbola are on the horizontal axis: the distance between the vertices is 2a. the distance between the centre of the hyperbola and each vertex is \|a\|. there exists points on the vertical axis such that the distance between each point and the centre of the hyperbola is \|b\|.	When the vertices of the hyperbola are on the vertical axis: the distance between the vertices is 2b. the distance between the centre of the hyperbola and each vertex is \|b\|. there exists points on the horizontal axis is such that the distance between each point and the centre is \|a\|.
As before, we can draw a rectangle of width 2a and height 2b in the graph, and use this rectangle to find that the slopes of the asymptotes are +b/a and -b/a.
Once you know the slopes of the asymptotes and centre of the hyperbola, sketch them on your graph paper. Then determine the locations of the vertices and go ahead and sketch the hyperbola that passes through the vertices and approaches the asymptotes. This method is more closely examined in the questions below.