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






11 and 12 Grade Math 

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


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Days of the Week | 7 Days of the Week | What are the Seven Days?

    Nov 30, 23 10:59 PM

    Days of the Weeks
    We know that, seven days of a week are Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. A day has 24 hours. There are 52 weeks in a year. Fill in the missing dates and answer the questi…

    Read More

  2. Types of Lines |Straight Lines|Curved Lines|Horizontal Lines| Vertical

    Nov 30, 23 01:08 PM

    Types of Lines
    What are the different types of lines? There are two different kinds of lines. (i) Straight line and (ii) Curved line. There are three different types of straight lines. (i) Horizontal lines, (ii) Ver…

    Read More

  3. 1 to 10 Times Tables | 1 - 10 Times Table Chart |Multiplication Tables

    Nov 30, 23 01:26 AM

    1 to 10 Times Tables
    Memorizing 1 to 10 Times Tables are very important for mental math and quick calculations. Times Tables are used during multiplication and division. Let us learn all the times tables from 1 to 10 to i…

    Read More