Problem 34 - Algebra and Trigonometry

Question

If all conics are defined in terms of a fixed point and a fixed line, how can you tell one kind of conic from another?

Step-by-Step Solution

Answer

Identifying a conic section is based on its unique properties. A circle has a constant radius from its center to all points on it. An ellipse maintains a constant sum of distances from any point on it to two foci. A parabola is formed based on the equality of distance from a point to a focus and a directrix. A hyperbola has a constant difference in distances from a point on it to two foci.

Step 1: Identifying Circle

A circle is a special case of an ellipse where the fixed point, called the center, and all points on the circle are equidistant. This is based on the characteristic that the distances from any point on the circle to the center (fixed point) is always equal to the radius.

Step 2: Identifying Ellipse

An ellipse is a conic that contains two fixed points, known as foci. Any point on the ellipse maintains a constant sum of distances to these two foci. This property differentiates it from other conics.

Step 3: Identifying Parabola

A parabola is a set of all points that are equidistant from a fixed point known as the focus and a line called the directrix. That means the distance from any point on the parabola to the focus is equal to the perpendicular distance of the point to the directrix.

Step 4: Identifying Hyperbola

A hyperbola is formed by the difference of distances from two fixed points called foci. Each point on a hyperbola has a constant difference in its distances to the two foci. This constant difference distinguishes a hyperbola from the other conics.