Position of a Point with respect to the Ellipse

We will learn how to find the position of a point with respect to the ellipse.

The point P (x$_{1}$, y$_{1}$) lies outside, on or inside the ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 according as $\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ – 1 > 0, = or < 0.

Let P (x$_{1}$, y$_{1}$) be any point on the plane of the ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 ………………….. (i)

From the point P (x$_{1}$, y$_{1}$) draw PM perpendicular to XX' (i.e., x-axis) and meet the ellipse at Q.

According to the above graph we see that the point Q and P have the same abscissa. Therefore, the co-ordinates of Q are (x$_{1}$, y$_{2}$).

Since the point Q (x$_{1}$, y$_{2}$) lies on the ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1.

Therefore,

$\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{2}^{2}}{b^{2}}$ = 1        

$\frac{y_{2}^{2}}{b^{2}}$ = 1 - $\frac{x_{1}^{2}}{a^{2}}$ ………………….. (i)

Now, point P lies outside, on or inside the ellipse according as

PM >, = or < QM

i.e., according as y$_{1}$ >, = or < y$_{2}$

i.e., according as $\frac{y_{1}^{2}}{b^{2}}$ >, = or < $\frac{y_{2}^{2}}{b^{2}}$

i.e., according as $\frac{y_{1}^{2}}{b^{2}}$ >, = or < 1 - $\frac{x_{1}^{2}}{a^{2}}$, [Using (i)]

i.e., according as $\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ >, = or < 1

i.e., according as $\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ - 1 >, = or < 0

Therefore, the point

(i) P (x$_{1}$, y$_{1}$) lies outside the ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 if PM > QM

i.e., $\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ - 1 > 0.

(ii) P (x$_{1}$, y$_{1}$) lies on the ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 if PM = QM

i.e., $\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ - 1 = 0.

(ii) P (x$_{1}$, y$_{1}$) lies inside the ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 if PM < QM

i.e., $\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ - 1 < 0.

Hence, the point P(x$_{1}$, y$_{1}$) lies outside, on or inside the ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 according as x$\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$  - 1 >, = or < 0.

Note:

Suppose E$_{1}$ = $\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ - 1, then the point P(x$_{1}$, y$_{1}$) lies outside, on or inside the ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1 according as E$_{1}$ >, = or < 0.

Solved examples to find the position of the point (x$_{1}$, y$_{1}$) with respect to an ellipse $\frac{x^{2}}{a^{2}}$ + $\frac{y^{2}}{b^{2}}$ = 1:

1. Determine the position of the point (2, - 3) with respect to the ellipse $\frac{x^{2}}{9}$ + $\frac{y^{2}}{25}$ = 1.  

Solution:

We know that the point (x$_{1}$, y$_{1}$) lies outside, on or inside the ellipse

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

$\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ – 1 > , = or  < 0.

For the given problem we have,

$\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ - 1 = $\frac{2^{2}}{9}$ + $\frac{(-3)^{2}}{25}$ – 1 = $\frac{4}{9}$ + $\frac{9}{25}$ - 1 = - $\frac{44}{225}$ < 0.

Therefore, the point (2, - 3) lies inside the ellipse $\frac{x^{2}}{9}$ + $\frac{y^{2}}{25}$ = 1.

2. Determine the position of the point (3, - 4) with respect to the ellipse $\frac{x^{2}}{9}$ + $\frac{y^{2}}{16}$ = 1.  

Solution:

We know that the point (x$_{1}$, y$_{1}$) lies outside, on or inside the ellipse

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

$\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ - 1 > , = or  < 0.

For the given problem we have,

$\frac{x_{1}^{2}}{a^{2}}$ + $\frac{y_{1}^{2}}{b^{2}}$ - 1 = $\frac{3^{2}}{9}$ + $\frac{(-4)^{2}}{16}$ - 1 = $\frac{9}{9}$ + $\frac{16}{16}$ - 1 = 1 + 1 - 1 = 1 > 0.

Therefore, the point (3, - 4) lies outside the ellipse $\frac{x^{2}}{9}$ + $\frac{y^{2}}{16}$ = 1.  

● The Ellipse

• Definition of Ellipse
• Standard Equation of an Ellipse
• Two Foci and Two Directrices of the Ellipse
• Vertex of the Ellipse
• Centre of the Ellipse
• Major and Minor Axes of the Ellipse
• Latus Rectum of the Ellipse
• Position of a Point with respect to the Ellipse
• Ellipse Formulae
• Focal Distance of a Point on the Ellipse
• Problems on Ellipse

11 and 12 Grade Math 

From Position of a Point with respect to the Ellipse to HOME PAGE