Of the following four points, three are an equal distance...

Of the following four points, three are an equal distance from the point $P(2,4)$ and the line $y=-4 .$ (a) Identify which three, and (b) find any two additional points that satisfy these conditions. $Q(-10,9) \quad R(2+4 \sqrt{2}, 3) \quad S(10,4)$
$T(2-4 \sqrt{5}, 5)$

Key Concepts

Distance Formula

The distance formula is used to compute the distance between any two points in the coordinate plane using the square root of the sum of the squared differences in their coordinates. It is essential for quantifying how far apart points are, regardless of their location, and is derived from the Pythagorean theorem.

Distance from a Point to a Line

The distance from a point to a line is determined by measuring the perpendicular distance between them. For horizontal or vertical lines, this can often be simplified to the absolute difference in the relevant coordinate. This concept is crucial when comparing distances from a point to various geometric entities such as lines.

Parabola as a Locus

A parabola is defined as the set of all points that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). This definition is key for understanding problems where distances from a given point and a given line are equal, enabling one to identify or construct the locus of points forming a parabola.

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