Find an equation in standard form for the hyperbola with vertices at (0, ±4) and foci at (0, ±5).

Solution:

It is given that, the hyperbola with vertices at (0, ±4) and foci at (0, ±5).

Since vertices and foci lie on the y-axis, then the equation of the hyperbola is

(y²/b²) - (x²/a²) = 1

The vertices are at points (0,±4), then b = 4.

The foci are at points (0,±5), then c = 5.

So,

c² = b² + a²

Now substitute the values,

5² = 4² + a²

By transposing we get,

a² = 5² - 4²

a² = 25 - 16

a² = 9

Hence the equation of the hyperbola is,

(y²/16) - (x²/9) = 1

Therefore, an equation in standard form for the hyperbola is (y²/16) - (x²/9) = 1.

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Find an equation in standard form for the hyperbola with vertices at (0, ±4) and foci at (0, ±5).

Summary:

An equation in standard form for the hyperbola with vertices at (0, ±4) and foci at (0, ±5) is (y²/16) - (x²/9)= 1.