Given the hyperbola (x2/4) – (y2/9) = 1, find the equations for its asymptotes
A. x = (1/3)y, x = (–1/3)y
B. y = (3/2)x, y = (–3/2)x
C. y = (1/3)x, y = (–1/3)x
D. x = (3/2)y, x = (–3/2)y
If you can't recall the formula, just reason it out. As x,y get big, the equation just looks like
x^2/4 = y^2/9
9x^2 = 4y^2
3x = 2y
y = 3/2 x or -3/2 x
To find the equations of the asymptotes of a hyperbola, you need to divide the equation of the hyperbola by the constant of the squared terms. In this case, you need to divide both sides of the equation by 1 to simplify it.
Dividing the equation (x^2/4) – (y^2/9) = 1 by 1, we get:
(x^2/4)– (y^2/9) = 1
Now let's rewrite this equation in the form of (x^2/a^2) – (y^2/b^2) = 1, where a and b are the coefficients of the squared terms. Comparing this equation with the given equation, we can see that a^2 = 4 and b^2 = 9.
The equation of the asymptotes of the hyperbola can be written as x/a = ± y/b. Substituting the values of a and b, we get:
x/2 = ± y/3
Multiplying both sides of the equation by 2, we have:
x = ± (2/3)y
Therefore, the equations for the asymptotes of the hyperbola are:
x = (2/3)y and x = -(2/3)y
So, the correct answer is D. x = (3/2)y, x = (–3/2)y.