Position of a Point w.r.t. an Ellipse

Understanding the position of a point with respect to an ellipse is a fundamental concept in coordinate geometry. It helps in determining whether a point lies inside, on, or outside the ellipse. An ellipse is a set of all points in a plane, the sum of whose distances from two fixed points (foci) is constant.

Standard Equation of an Ellipse

The standard form of the equation of an ellipse with its center at the origin (0, 0) and major axis along the x-axis is given by:

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$

where:

• (a) is the semi-major axis length
• (b) is the semi-minor axis length

Position of a Point (x1, y1) w.r.t. an Ellipse

To determine the position of a point ((x_1, y_1)) with respect to the ellipse given by the equation (\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1), we substitute the coordinates of the point into the equation of the ellipse.

Let (S) be the result of the substitution:

$$ S = \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} $$

The position of the point is determined as follows:

Condition	Position of Point	Interpretation
(S = 1)	On the ellipse	The point lies on the boundary of the ellipse.
(S < 1)	Inside the ellipse	The point lies within the ellipse.
(S > 1)	Outside the ellipse	The point lies outside the ellipse.

Examples

Let's consider an ellipse with the equation (\frac{x^2}{16} + \frac{y^2}{9} = 1), where (a = 4) and (b = 3).

Example 1: Point on the Ellipse

Determine if the point (P(2, \sqrt{5})) lies inside, on, or outside the ellipse.

Substitute (x_1 = 2) and (y_1 = \sqrt{5}) into the equation:

$$ S = \frac{2^2}{16} + \frac{(\sqrt{5})^2}{9} = \frac{1}{4} + \frac{5}{9} = \frac{9}{36} + \frac{20}{36} = \frac{29}{36} $$

Since (S < 1), the point (P(2, \sqrt{5})) lies inside the ellipse.

Example 2: Point Outside the Ellipse

Determine if the point (Q(5, 0)) lies inside, on, or outside the ellipse.

Substitute (x_1 = 5) and (y_1 = 0) into the equation:

$$ S = \frac{5^2}{16} + \frac{0^2}{9} = \frac{25}{16} $$

Since (S > 1), the point (Q(5, 0)) lies outside the ellipse.

Example 3: Point on the Ellipse

Determine if the point (R(4, 0)) lies inside, on, or outside the ellipse.

Substitute (x_1 = 4) and (y_1 = 0) into the equation:

$$ S = \frac{4^2}{16} + \frac{0^2}{9} = 1 $$

Since (S = 1), the point (R(4, 0)) lies on the ellipse.

Conclusion

The position of a point with respect to an ellipse can be easily determined by substituting the point's coordinates into the ellipse's equation and comparing the result with 1. This method is essential in problems involving ellipses in coordinate geometry and has applications in various fields such as astronomy, physics, and engineering design.