__Category__: Analytic Geometry, Algebra"Published in Newark, California, USA"

Find the equation of hyperbola if the asymptotes are x + y = 1 and x - y = 1, and passing through through the point (3, 1).

__Solution__:

The first thing that we have to do is to get the point of intersection of the asymptotes as follows

x + y = 1 x + y = 1

--------------->

x - y = 1 x - y = 1

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2x = 2

x = 1

Substitute the value of x to either one of the asymptotes to get the value of y, we have

x + y = 1

1 + y = 1

y = 0

The center of the hyperbola is C(1, 0).

To illustrate the problem, it is better if you sketch the graph of the two asymptotes and a point as follows

Photo by Math Principles in Everyday Life |

Next, we need to get the values of a and b which are the values of semi-transverse axis and semi-conjugate axis of hyperbola. Since the slopes of two asymptotes of the hyperbola are +1 and -1, we can solve for the values of a and b as follows

Consider the positive sign in getting the values of a and b and the above equation becomes

The equation of the hyperbola in standard form if the transverse axis is parallel to x-axis is

But C(1, 0) and a = b, the above equation becomes

To solve for the value of a, substitute the values of x and y from the given point P(3, 1) as follows

Therefore, the equation of hyperbola in standard form is

We can also express the equation of hyperbola in general form as follows

Multiply both sides of the equation by 3, we have