Find the standard and general equation of a hyperbola whose center at the origin. And also find the equation of the asymptotes. a) b = 1, a = 5, transverse axis at y-axis
Question
Find the standard and general equation of a hyperbola whose center at the origin. And also find the equation of the asymptotes. a) b = 1, a = 5, transverse axis at y-axis
Solution
The standard equation of a hyperbola with its center at the origin depends on the orientation of the transverse axis.
a) If the transverse axis is along the y-axis, the standard equation of the hyperbola is:
(y^2/a^2) - (x^2/b^2) = 1
Given that a = 5 and b = 1, the equation becomes:
(y^2/25) - (x^2/1) = 1
The general form of the equation of a hyperbola is obtained by multiplying through by the common denominator (in this case, 25):
y^2 - 25x^2 = 25
The equations of the asymptotes of the hyperbola are given by:
y = ± (a/b)x
Substituting the given values of a and b, we get:
y = ± 5x
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