Position of a Point with respect to a Circle

Position of a Point with respect to a Circle

Let thecircle be x²+y² + 2gx +2fy + c = 0 Point P lies outside on, or inside thecircle according to as CP is greater than, equal to or less than radius,respectively I.e.

\(\sqrt{{{({{x}_{1}}+h)}^{2}}+{{({{y}_{1}}+f)}^{2}}}>,=,<\sqrt{{{g}^{2}}+{{f}^{2}}-c}\).

S₁ = x₁² + y₁² +2gx₁ + 2fy₁ + c = >, =, < 0

Maximum and Minimum Distance of a Pointfrom the Circle: Letany point p ≡ (x₁, y₁) and a circle.

S = x² + y² +2gx + 2fy + c = 0

The center and radius of the circle are (-g, -f) and \(\sqrt{{{g}^{2}}+{{f}^{2}}-c}\) Respectively

The maximum and minimum distance from P (x₁, y₁) to the circle, respectively, are

PB = CP + PC = r + PC

and PA = |CP – CA| = |PC – r| (P is outside or inside)

where r = \(\sqrt{{{g}^{2}}+{{f}^{2}}-c}\).

Example: Find Position of a point with respect to a circle, point P (10, 7) from the x² + y² – 4x – 2y – 20 = 0.

Solution: Given that

x² + y² – 4x – 2y – 20 = 0.

Since S₁ = x₁² + y₁² – 4x₁ – 2y₁ – 20 = 0. at given point

S₁ = (10)² + (7)² – 4(10) – 2(7) – 20 = 0

S₁ > 0, P lines on the outside circle.