Find an equation in standard form for the hyperbola with vertices at (0, ±2) and foci at (0, ±7)?

Solution:

Given:

Vertices at (0, ±2) and foci at (0, ±7)

As both vertices and foci lie on the y-axis

Equation of hyperbola is y²/b² - x²/a² = 1

Vertices = (0, ±2)

b = 2

Foci = (0, ±7)

be = 7

Eccentricity e = 7/2

We know that,

e = √(1 + a²/b²)

Substituting the values

7/2 = √(1 + a²/b²)

Squaring on both sides

49/4 = 1 + a²/b²

⇒ a²/b² = 49/4 - 1

⇒ a^2/b² = 45/4

Here,

a² = 45/4 × 2²

⇒ a² = 45/4 × 4

⇒ a² = 45

⇒ a = 3√5

So the equation of hyperbola is

y²/4 - x²/45 = 1

Therefore, the equation of hyperbola in standard form is y²/4 - x²/45 = 1.

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Find an equation in standard form for the hyperbola with vertices at (0, ±2) and foci at (0, ±7)?

Summary:

An equation in standard form for the hyperbola with vertices at (0, ±2) and foci at (0, ±7) is y²/4 - x²/45 = 1.