Can a Parabola Exist? Using Distance Formula with Point, Focus, and Directrix

Understanding Parabola Definition

• A parabola is defined as the set of all points equidistant from a focus point and a directrix.
• To determine if a given point P can exist on a parabola, we must check if its distance to the focus equals its distance to the directrix.

Calculating Distances

• The given point P is at (6, 14), the focus F is at (-8, 2), and the directrix is the line y = -6.
• The vertical distance from point P (y=14) to the directrix (y=-6) is calculated as 14 - (-6) = 20.
• The distance from point P (6, 14) to the focus F (-8, 2) is calculated using the distance formula: √((6 - (-8))^2 + (14 - 2)^2).

Determining Parabola Existence

• The distance from P to F is √((6 + 8)^2 + (12)^2) = √(14^2 + 12^2) = √(196 + 144) = √340.
• Since √340 is not equal to 20 (as 20^2 = 400), the distance from P to F is not equal to the distance from P to the directrix.
• ❌ Therefore, the point P (6, 14) cannot exist on a parabola with the focus at (-8, 2) and the directrix at y = -6.