Question

Write the equation of the hyperbola centered at the origin, with length of the horizontal transverse axis 6 and the curve passes through the point (9,−8).

Answer 1

The equation of a hyperbola with a horizontal transverse axis of length 6 and passing through the point (9, -8) is x^2/9 - y^2/8 = 1.

Step-by-step explanation:

To write the equation of the hyperbola centered at the origin with a horizontal transverse axis of length 6 and passing through the point (9,−8), we first note the transverse axis length provides the distance between the two vertices on the horizontal axis (x-axis), which is 2a. Therefore, the distance from the center to each vertex, 'a', is half of this, or 3 units. The standard form of a horizontal hyperbola centered at the origin is (x2/a2) - (y2/b2) = 1.

Since the point (9,−8) lies on the hyperbola, we can substitute x = 9 and y = −8 into the equation to find 'b'. After substitution, we get (92/32) - (82/b2) = 1, simplifying to 9 - (64/b2) = 1. Solving this for b2 gives us b2 = 64/8, thus b2 = 8.

Finally, the equation of the hyperbola is x2/9 - y2/8 = 1.

Answer 2

To find the equation of a hyperbola, centered at the origin with a horizontal transverse axis of length 6 that passes through the point (9, -8), we use the standard form x²/a² - y²/b² = 1. The transverse axis length gives us a = 3. By substituting the given point into the equation, we solve for b and get the final equation: x²/9 - y²/32 = 1.

Step-by-step explanation:

To write the equation of a hyperbola centered at the origin with a given transverse axis and a point it passes through, we need to understand the standard form of a hyperbola's equation and use the given information to find the specific values. For a horizontal hyperbola centered at the origin, the equation is x²/a² - y²/b² = 1, where 2a is the length of the transverse axis and 2b is the length of the conjugate axis. With the transverse axis 6 units long, we have a = 3. The hyperbola passing through the point (9, -8) must satisfy the equation, so substituting x = 9 and y = -8 into the equation gives us 9²/a² - (-8)²/b² = 1. We can solve this equation to find the value of b, and once we have both a and b, we can write the entire equation of the hyperbola.

First, we have 9²/3² - 8²/b² = 1 which simplifies to 3 - 64/b² = 1. Solving for b² gives us b² = 64/(3 - 1) = 32. Therefore, b = √32. The final equation of the hyperbola is x²/9 - y²/32 = 1.