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A hedge is to be constructed in the shape of a hyperbola near a fountain at the center of a yard. The hedge will follow the asymptotes y=3x and y=-3x. What is the equation of the hyperbola in standard form? a) y = 3/x b) xy = 3 c) xy = -3 d) y = 1/(3x)

Answer :

Final answer:

The equation of the hyperbola in standard form is 9x² - y² = 1. None of above given options in question are correct.

Explanation:

To find the equation of the hyperbola in standard form when the asymptotes are given by y = 3x and y = -3x, we can follow these steps:

Start with the general equation for a hyperbola with asymptotes parallel to the coordinate axes:

(x / a)² - (y / b)² = 1

We know that the asymptotes are y = 3x and y = -3x. Asymptotes are lines of the form y = mx, where m is the slope of the asymptotes. In this case, m = 3 and m = -3.

The slope of the asymptotes (m) is related to the parameters a and b in the hyperbola's standard form as follows:

a = |1/m|

b = |a × m|

Using the slope m = 3 for the first asymptote, we can find the value of a:

a = |1/3| = 1/3

Using the slope m = -3 for the second asymptote, we can find the value of b:

b = |(1/3) × (-3)| = 1

Now, we have the values of a and b. Substitute these into the standard form of the hyperbola:

(x / (1/3))² - (y / 1)² = 1

Simplify:

(3x)² - y² = 1

Further simplify:

9x² - y² = 1

So, the equation of the hyperbola in standard form is 9x² - y² = 1.

The correct answer is not one of the provided options (a) y = 3/x, (b) xy = 3, (c) xy = -3, or (d) y = 1/(3x). The equation is 9x² - y² = 1

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