Therefore, the equation of the hyperbola with a given origin has vertices at (0, ±9) and foci at (0, ±10) is [tex]\frac{x^{2} }{81}-\frac{y^{2} }{19} =1[/tex].
Given, that a hyperbola centred at the origin has vertices at (0, ±9) and foci at (0, ±10).
In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows.
The formula for a hyperbola centred at the origin is [tex]\frac{(x-h)^{2} }{a^{2} } -\frac{(y-k)^{2} }{b^{2} } =1[/tex]
Where (h, k) is the center = (0, 0)
Distance from centre to vertices a = 9 ⇒ a² = 81
Distance from centre to vertices which is given from the foci c = 10
⇒ c² = 100
Using the Pythagorean formula, c²= a²+ b²
Substituting the values 100 = 81 + b²
So we get, b²= 100 - 81 = 19
Substituting the values in the standard form [tex]\frac{x^{2} }{81}-\frac{y^{2} }{19} =1[/tex].
Therefore, the equation of the hyperbola with a given origin has vertices at (0, ±9) and foci at (0, ±10) is [tex]\frac{x^{2} }{81}-\frac{y^{2} }{19} =1[/tex].
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