17Calculus - Hyperbolas in Rectangular Coordinates

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On this page, we discuss hyperbolas in rectangular coordinates. For a discussion of hyperbolas in polar form, see this separate page.

Hyperbola

The hyperbola is the most complicated of the three and, consequently, the most interesting. Figure 1 contains information about the hyperbola.
The vertices ($$\pm a$$ on this plot) and the foci (F1 and F2 on this plot) lie on the transverse axis with the center at $$(h,k)$$. The standard equations are

 $$\displaystyle{ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 }$$ horizontal transverse axis $$\displaystyle{ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 }$$ vertical transverse axis

As we did with the ellipse, we define an intermediate value $$c^2=a^2+b^2$$ which will help us locate the foci.
For a hyperbola, we need to know the equations of the lines in blue on the plot. These are asymptotes.
For a horizontal transverse axis, the asymptotes are $$\displaystyle{ y=k \pm \frac{b}{a}(x-h) }$$.
For a vertical transverse axis, the asymptotes are $$\displaystyle{ y=k \pm \frac{a}{b}(x-h) }$$.
Similar to the ellipse, we define the eccentricity as $$e=c/a$$. The results are summarized next.

$$\displaystyle{ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 }$$

horizontal transverse axis

center

$$(h,k)$$

vertices

$$(h \pm a, k)$$

foci

$$(h \pm c,k)$$

asymptotes

$$\displaystyle{ y=k \pm \frac{b}{a}(x-h) }$$

$$c^2=a^2+b^2$$

eccentricity

$$e=c/a$$

$$\displaystyle{ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 }$$

vertical transverse axis

center

$$(h,k)$$

vertices

$$(h, k \pm a)$$

foci

$$(h,k \pm c)$$

asymptotes

$$\displaystyle{ y=k \pm \frac{a}{b}(x-h) }$$

$$c^2=a^2+b^2$$

eccentricity

$$e=c/a$$

 Notes

1. Notice that the value of c is different here than for an ellipse.
2. Since $$c > a$$, $$e > 1$$.

The hyperbola is quite a complicated graph with lots of features. Figure 2 gives you an idea of some of the other features that are involved with hyperbolas. Here is a video that goes into more detail.

MIP4U - Conic Sections: The Hyperbola part 1 of 2 [4min-12secs]

video by MIP4U

Okay, time for some practice problems on hyperbolas. Although it may seem like we have given you a lot of information on this page, we have just skimmed the surface in the discussion of these three figures, especially the hyperbola. There are a lot more interesting features that we hope you get to explore in your class.

Practice

Sketch the graph of the hyperbola $$\displaystyle{ (y+4)^2 - \frac{x^2}{25} = 1 }$$.

Problem Statement

Sketch the graph of the hyperbola $$\displaystyle{ (y+4)^2 - \frac{x^2}{25} = 1 }$$.

Solution

PatrickJMT - 1592 video solution

video by PatrickJMT

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Graph $$\displaystyle{ \frac{x^2}{49} - \frac{y^2}{25} = 1 }$$.

Problem Statement

Graph $$\displaystyle{ \frac{x^2}{49} - \frac{y^2}{25} = 1 }$$.

Solution

PatrickJMT - 1604 video solution

video by PatrickJMT

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Write the equation of the hyperbola that has vertices $$(-2,-5), (4,-5)$$ and foci $$(-4,-5), (6,-5)$$.

Problem Statement

Write the equation of the hyperbola that has vertices $$(-2,-5), (4,-5)$$ and foci $$(-4,-5), (6,-5)$$.

Solution

PatrickJMT - 1605 video solution

video by PatrickJMT

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Graph $$\displaystyle{ \frac{(x-2)^2}{4} - \frac{(y+3)^2}{9} = 1 }$$.

Problem Statement

Graph $$\displaystyle{ \frac{(x-2)^2}{4} - \frac{(y+3)^2}{9} = 1 }$$.

Solution

MIP4U - 1606 video solution

video by MIP4U

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Graph $$\displaystyle{ \frac{(y+4)^2}{4} - \frac{(x-2)^2}{16} = 1 }$$.

Problem Statement

Graph $$\displaystyle{ \frac{(y+4)^2}{4} - \frac{(x-2)^2}{16} = 1 }$$.

Solution

MIP4U - 1607 video solution

video by MIP4U

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