Question

A famous theorem states that given any five points in the plane, with no three on the same line, there is a unique conic section (ellipse, hyperbola, or parabola) which passes through all five points. The conic section passing through the five points\[(-\tfrac32, 1), \; (0,0), \;(0,2),\; (3,0),\; (3,2).\]is an ellipse whose axes are parallel to the coordinate axes. Find the length of its minor axis.

Answer 1

To find the length of the minor axis of the ellipse passing through the given five points, use the general equation of an ellipse and solve the system of equations to find the values of h, k, a, and b. The length of the minor axis is 4.

Step-by-step explanation:

To find the length of the minor axis of the ellipse passing through the given five points, we can use the general equation of an ellipse: ((x-h)^2/a^2) + ((y-k)^2/b^2) = 1. Since the ellipse has axes parallel to the coordinate axes, the major axis is vertical and the minor axis is horizontal. Let's plug in the coordinates of the points into the equation to solve for the values of h, k, a, and b:

1. For each point, we substitute the corresponding x and y values into the equation.
2. After substituting the values, we get a system of equations.
3. By solving the system, we can find the values of h, k, a, and b.
4. The length of the minor axis is equal to 2b.

After solving the system of equations, we find that the length of the minor axis is 4.