Graph the conic section of 9x^2-36y^2=324

Answer 1

To graph the conic section 9x^2-36y^2=324, we can start by rearranging it into a standard form:

9x^2/324 - 36y^2/324 = 1

x^2/36 - y^2/9 = 1

This is the equation of a hyperbola with a horizontal transverse axis.

To graph it, we can plot the center (0,0), which is the midpoint of the two branches of the hyperbola. Then, we can locate the vertices, which are a distance of 6 units to the right and left of the center, giving us the points (6,0) and (-6,0). We can also locate the foci, which are a distance of sqrt(36+9) = 3sqrt(5) units to the right and left of the center, giving us the points (3sqrt(5),0) and (-3sqrt(5),0).

Finally, we can sketch the two branches of the hyperbola, which are asymptotic to the lines y = 3x/2 and y = -3x/2 as x becomes infinitely large in either direction. The resulting graph should look like this: