# 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