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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
-----------------
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
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| 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
