Question

Find the equation of the hyperbola satisfying the given conditions: Vertices (0,±5), foci (0,±8)​

Answer 1

⠀⠀ıllıllı uoᴉʇnloS ıllıllı

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Vertices (0, ± 5), foci (0, ± 8)

Here, the vertices are on the y-axis.

Therefore:

The equation of the hyperbola is of the form

• y²/a² - x²/b² = 1

Since, the vertices are (0, ± 5), a = 5.

Since, the foci are (0, ± 8), c = 8.

We know that:

• a² + b² = c²

Therefore:

5² + b² = 8²

b² = 64 - 25

= 39

• Thus, the equation of the hyperbola is y²/25 - x²/39 = 1

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Answer 2

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Given that: Foci (0, ±13), Conjugate axis length = 24

It is noted that the foci are on the y-axis.

Therefore, the equation of the hyperbola is of the form:

(y^2/a^2)-(x^2/b^2) = 1 …(1)

Since the foci are (0, ±13), we can get

C = 13

It is given that, the length of the conjugate axis is 24,

It becomes 2b = 24

b= 24/2

b= 12

And, we know that a^2 + b^2 = c^2

To find a, substitute the value of b and c in the above equation:

a^2 + 5^2 = 8^2

a^2 = 64-25

a^2 = 39

Now, substitute the value of a and b in equation (1), we get

(y^2/25)-(x^2/39) = 1, which is the required equation of the hyperbola

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Hope it's Helpful.....:)