Find the Equation. Please check the answers and help. Thanks!

1.) Ellipse with center (0,0), foci on x-axis; x intercepts; major axis of length 12, minor acis of length 8.

I got : x^2/144 + y^2/64 = 1

2.) A parabola with vertex (0, 0) and focus (5, 0).

I got : y^2 = 20x

3.) Hyperbole with center at (0, 0), vertex (2, 0); passing through (4, 31/2 )

I don't know this one please help, confused with center and vertex and don't know what passing through means.

last one is 3 and a half NOT 31/2

1.) Ellipse with center (0,0), foci on the x-axis; x-intercepts; major axis of length 12, minor axis of length 8.

To find the equation of the ellipse, we need to know the properties of the ellipse, such as its center, foci, and lengths of the major and minor axes.

In this case, we know that the center of the ellipse is (0,0) and the foci lie on the x-axis. The major axis has a length of 12 and the minor axis has a length of 8.

The general equation of an ellipse centered at the origin is given by:

(x^2 / a^2) + (y^2 / b^2) = 1,

where "a" represents the semi-major axis (half the length of the major axis) and "b" represents the semi-minor axis (half the length of the minor axis).

Since the center is (0,0), we have a = 12/2 = 6 for the semi-major axis length and b = 8/2 = 4 for the semi-minor axis length.

Substituting these values into the equation, we get:

(x^2 / 36) + (y^2 / 16) = 1.

So, the equation of the ellipse is x^2/36 + y^2/16 = 1.

2.) A parabola with vertex (0,0) and focus (5,0).

To find the equation of the parabola, we need to know the properties of the parabola, such as its vertex and focus.

In this case, we know that the vertex is (0,0) and the focus is (5,0).

The general equation of a parabola with vertex at the origin and focus at (a,0) is given by:

y^2 = 4ax.

Since the vertex is at (0,0) and the focus is at (5,0), we can substitute the values into the equation as follows:

y^2 = 4 * 5 * x.

Simplifying, we get:

y^2 = 20x.

So, the equation of the parabola is y^2 = 20x.

3.) Hyperbola with center at (0,0), vertex (2,0); passing through (4, 31/2).

To find the equation of the hyperbola, we need to know the properties of the hyperbola, such as its center, vertices, and other point(s) on the hyperbola.

In this case, we know that the center is (0,0) and one vertex is (2,0). We are also given another point (4, 31/2) that lies on the hyperbola.

Using the information given, we can determine the equation of the hyperbola.

The general equation of a hyperbola centered at the origin is:

(x^2 / a^2) - (y^2 / b^2) = 1,

where "a" represents the distance from the center to the vertex and "b" represents the distance from the center to the foci.

Since the center is at (0,0) and the vertex is at (2,0), we have a = 2 for the distance from the center to the vertex.

To find "b," we can use the distance formula. The distance from the center (0,0) to the point (4, 31/2) is:

√((4-0)^2 + (31/2-0)^2) = √(16 + 961/4) = √(256/4 + 961/4) = √(1217/4).

So, b is equal to half this distance:

b = (1/2) √(1217/4) = √(1217)/4.

Therefore, the equation of the hyperbola is:

(x^2 / 4) - (y^2 / (√(1217)/4)^2) = 1,

simplifying:

x^2 - (y^2 / (1217/16)) = 1.

So, the equation of the hyperbola is x^2 - (16y^2 / 1217) = 1.