To determine the standard-form equation, we'll have to complete the square for both x and y. It will be really helpful to re-group our terms to do that:

Adding 9 will complete the square for x, since

Adding 16 will complete the square for y, since

Now we just need to simplify. Re-write the left side as two binomials squared, and add the numbers on the right side:

Recall that the standard from for the equation of a circle is

where (h, k) is the center, and r is the radius. We are given the center (3, -4) and radius 2. Therefore, h = 3, k = -4, and r = 2. Plugging these vaules into the equation gives us

The coefficient of the squared term tells us whether the parabola faces up or down. Parabolas in general, as in the parent function, are in the shape of a U. In the equation given, the coefficient of the squared term is . Generally, if the coefficient of the squared term is positive, the parabola faces up. If the coefficient is negative, the parabola faces down. Since is negative, our parabola must face down.

This equation should be thought of as .

This means that the hyperbola will be determined by a box with x-intercepts at and y-intercepts at .

The hyperbola was incorrectly drawn with the intercepts at instead.

The equation for the eccentricity of an ellipse is , where is eccentricity, is the distance from the foci to the center, and is the square root of the larger of our two denominators.

Our denominators are and , so .

To find , we must use the equation , where is the square root of the smaller of our two denominators.

This gives us , so .

Therefore, we can see that

.

Since the circle is centered at we use the most basic form for the equation for a circle:

.

Given the circle has a radius of marks, which represent units each, the circle has a radius of units.

We then plug in for :

and simplify: .

To determine the standard-form equation, we'll have to complete the square for both x and y. It will be really helpful to re-group our terms to do that:

Adding 9 will complete the square for x, since

Adding 16 will complete the square for y, since

Now we just need to simplify. Re-write the left side as two binomials squared, and add the numbers on the right side:

Recall that the standard from for the equation of a circle is

where (h, k) is the center, and r is the radius. We are given the center (3, -4) and radius 2. Therefore, h = 3, k = -4, and r = 2. Plugging these vaules into the equation gives us

This equation should be thought of as .

This means that the hyperbola will be determined by a box with x-intercepts at and y-intercepts at .

The hyperbola was incorrectly drawn with the intercepts at instead.

This equation should be thought of as .

This means that the hyperbola will be determined by a box with x-intercepts at and y-intercepts at .

The hyperbola was incorrectly drawn with the intercepts at instead.