Conic Sections: Introduction to Conic Sections

Distance Formula

We will begin our discussion of conic sections by reviewing the Distance Formula and the Midpoint Formula. 

Conic sections can be studied as figures on a coordinate plane. They can be described in terms of the distances between points. This is why the distance formula is an integral part in the study of conic sections.

Analyze the following diagram.

Two triangles, one describing the Pythagorean Theorem and one using the Pythagorean Theorem to find the Distance Formula between points (x 1, y 1) and (x 2, y 2).

On the left you have a triangle with sides a, b, and c. Side c is the hypotenuse and is equal to square root of quantity a squared plus b squared.

On the right you have the same triangle with points A and B defined. A has coordinate (x1, y1) and B has coordinate (x2, y2).

Using the Pythagorean Theorem again we can find side d, which is the distance between points A and B.

d equals the square root of the quantity x 2 minus x 1 quantity squared plus y 2 minus y 1 quantity squared

This is the definition of the distance between two points.