Question

Find the equation of the hyperbola with vertices at (0,−2) and (0,2), and foci at (0,−3) and (0,3). Use the equation to select the coordinates among the options for the point on that hyperbola. Note that some of the coordinates are approximate.

a. (4.123,5)
b. (2.5,3)
c. (5.325,6)
d. (2.873,4)

Answer 1

The equation of the hyperbola is
`\((y^2)/(4) - (x^2)/(5) = 1\)`. Among the given options, the point (2.5, 3) lies on this hyperbola. Option B is correct.

The standard form equation of a hyperbola with a vertical axis is given by:

`\[ ((y - k)^2)/(a^2) - ((x - h)^2)/(b^2) = 1 \]`

where (h, k) is the center, a is the distance from the center to the vertices, and b is the distance from the center to the foci.

Given the vertices at (0, -2) and (0, 2), we can see that the center is at the origin (0,0). The distance from the center to the vertices is (a = 2).

The distance from the center to the foci is (c = 3).

The relationship between (a), (b), and (c) in a hyperbola is given by
`\(c^2 = a^2 + b^2\)`. Therefore,
`\(b^2 = c^2 - a^2 = 3^2 - 2^2 = 5\)`.

Now, plug these values into the equation:

`\[ (y^2)/(2^2) - (x^2)/(√(5)^2) = 1 \]`

Simplifying, we get:

`\[ (y^2)/(4) - (x^2)/(5) = 1 \]`

Now, let's check which point from the given options lies on this hyperbola.

a.
`\((4.123,5)\):`

`\[ (5^2)/(4) - (4.123^2)/(5) \approx 1 \]`(approximately)

b.
`\((2.5,3)\)`:

`\[ (3^2)/(4) - (2.5^2)/(5) = 1 \]`

c.
`\((5.325,6)\)`:

`\[ (6^2)/(4) - (5.325^2)/(5) \approx 1 \]`(approximately)

d.
`\((2.873,4)\)`:

`\[ (4^2)/(4) - (2.873^2)/(5) \approx 1 \]` (approximately)

So, the correct answer is (b)
`\((2.5,3)\)`.