Vertex of the Ellipse

We will discuss about the vertex of the ellipse along with the examples.


Definition of the vertex of the ellipse:

The vertex is the point of intersection of the line perpendicular to the directrix which passes through the focus cuts the ellipse.

Suppose the equation of the ellipse be \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 then, from the above figure we observe that the line perpendicular to the directrix KZ and passing through the focus S cuts the ellipse at A and A'.

The points A and A', where the ellipse meets the line joining the foci S and S' are called the vertices of the ellipse.

Therefore, the ellipse has two vertices A and A' whose co-ordinates are (a, 0) and (- a, 0) respectively.

Solved examples to find the vertex of an ellipse:

1. Find the coordinates of the vertices of the ellipse 9x\(^{2}\) + 16y\(^{2}\) - 144 = 0.

Solution:

The given equation of the ellipse is 9x\(^{2}\) + 16y\(^{2}\) - 144 = 0

Now form the above equation we get,

9x\(^{2}\) + 16y\(^{2}\) = 144

Dividing both sides by 144, we get

\(\frac{x^{2}}{16}\) + \(\frac{y^{2}}{9}\) = 1

This is the form of \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1, (a\(^{2}\) > b\(^{2}\)), where a\(^{2}\) = 16 or a = 4 and b\(^{2}\) = 9 or b = 3

We know the coordinates of the vertices are (a, 0) and (-a, 0).

Therefore, the coordinates of the vertices of the ellipse 9x\(^{2}\) + 16y\(^{2}\) - 144 = 0 are (4, 0) and (-4, 0).

 

2. Find the coordinates of the vertices of the ellipse 9x\(^{2}\) + 25y\(^{2}\) - 225 = 0.

Solution:

The given equation of the ellipse is 9x\(^{2}\) + 25y\(^{2}\) - 225 = 0

Now form the above equation we get,

9x\(^{2}\) + 25y\(^{2}\) = 225

Dividing both sides by 225, we get

\(\frac{x^{2}}{25}\) + \(\frac{y^{2}}{9}\) = 1

Comparing the equation \(\frac{x^{2}}{25}\) + \(\frac{y^{2}}{9}\) = 1

with the standard equation of ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 (a\(^{2}\) > b\(^{2}\)) we get,

a\(^{2}\) = 25 or a = 5 and b\(^{2}\) = 9 or b = 3

We know the coordinates of the vertices are (a, 0) and (-a, 0).

Therefore, the coordinates of the vertices of the ellipse 9x\(^{2}\) + 25y\(^{2}\) - 225 = 0 are (5, 0) and (-5, 0).

● The Ellipse





11 and 12 Grade Math 

From Vertex of the Ellipse to HOME PAGE




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.



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.




Share this page: What’s this?

Recent Articles

  1. Addition of Decimals | How to Add Decimals? | Adding Decimals|Addition

    Apr 24, 25 01:45 AM

    Addition of Decimals
    We will discuss here about the addition of decimals. Decimals are added in the same way as we add ordinary numbers. We arrange the digits in columns and then add as required. Let us consider some

    Read More

  2. Addition of Like Fractions | Examples | Videos | Worksheet | Fractions

    Apr 23, 25 09:23 AM

    Adding Like Fractions
    To add two or more like fractions we simplify add their numerators. The denominator remains same. Thus, to add the fractions with the same denominator, we simply add their numerators and write the com…

    Read More

  3. Subtraction | How to Subtract 2-digit, 3-digit, 4-digit Numbers?|Steps

    Apr 23, 25 12:41 AM

    Subtraction Example
    The answer of a subtraction sum is called DIFFERENCE. How to subtract 2-digit numbers? Steps are shown to subtract 2-digit numbers.

    Read More

  4. Subtraction of 4-Digit Numbers | Subtract Numbers with Four Digit

    Apr 23, 25 12:38 AM

    Properties of Subtraction of 4-Digit Numbers
    We will learn about the subtraction of 4-digit numbers (without borrowing and with borrowing). We know when one number is subtracted from another number the result obtained is called the difference.

    Read More

  5. Subtraction with Regrouping | 4-Digit, 5-Digit and 6-Digit Subtraction

    Apr 23, 25 12:34 AM

     Subtraction of 5-Digit Numbers with Regrouping
    We will learn subtraction 4-digit, 5-digit and 6-digit numbers with regrouping. Subtraction of 4-digit numbers can be done in the same way as we do subtraction of smaller numbers. We first arrange the…

    Read More