Write the equation of the hyperbola for which the length of the transverse axis is 6 units long anf the foci are at (0,5) and (0,-5).
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The length of transverse axis is 2a
To find the equation of the hyperbola, we need to use the given information about the length of the transverse axis and the coordinates of the foci.
Let's start by considering the standard form of a hyperbola centered at the origin:
(x^2 / a^2) - (y^2 / b^2) = 1
Where "a" represents the distance from the center to the vertex along the transverse axis, and "b" represents the distance from the center to the vertex along the conjugate axis.
In this case, the length of the transverse axis is given as 6 units, which means the value of "a" is 6/2 = 3 units.
We also know that the distance from the center to the foci is 5 units, so the value of "c" (the distance from the center to the foci) is 5 units.
Now we can use the relationship between "a", "b", and "c" in a hyperbola, which is given by the equation: c^2 = a^2 + b^2.
Substituting the given values:
5^2 = 3^2 + b^2
25 = 9 + b^2
b^2 = 25 - 9
b^2 = 16
b = 4
Now we have values for "a" and "b", so we can substitute them into the standard form equation:
(x^2 / 3^2) - (y^2 / 4^2) = 1
Simplifying, we have:
(x^2 / 9) - (y^2 / 16) = 1
This is the equation of the hyperbola with a transverse axis length of 6 units and foci at (0,5) and (0,-5).