Answer:
D. 25x^2-9y^2=225
Step-by-step explanation:
D. 25x^2-9y^2=225
The equation of the hyperbola centered at the origin that satisfies the given conditions is [tex]\rm \dfrac{x^2}{9}-\dfrac{y^2}{25}=1[/tex].
The hyperbola has x-intercepts, so it has a horizontal transverse axis.
The standard form of the equation of a hyperbola with a horizontal transverse axis is;
[tex]\rm \dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1[/tex]
The center is at (h,k).
The distance between the vertices is 2a.
The value of a and b are;
For 'a': 2a = x₂ - x₁ = 3 - (-3) = 3 + 3 = 6
a = 6/2 = 3 and b =5.
The equation of the hyperbola centered at the origin that satisfies the given conditions is;
[tex]\rm \dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1\\\\\rm \dfrac{(x-0)^2}{3^2}-\dfrac{(y-0)^2}{5^2}=1\\\\\rm \dfrac{x^2}{9}-\dfrac{y^2}{25}=1[/tex]
Hence, the equation of the hyperbola centered at the origin that satisfies the given conditions is [tex]\rm \dfrac{x^2}{9}-\dfrac{y^2}{25}=1[/tex].
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