# 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

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.

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

## Recent Articles

1. ### Lines of Symmetry | Symmetry of Geometrical Figures | List of Examples

Aug 10, 24 04:59 PM

Learn about lines of symmetry in different geometrical shapes. It is not necessary that all the figures possess a line or lines of symmetry in different figures.

2. ### Symmetrical Shapes | One, Two, Three, Four & Many-line Symmetry

Aug 10, 24 02:25 AM

Symmetrical shapes are discussed here in this topic. Any object or shape which can be cut in two equal halves in such a way that both the parts are exactly the same is called symmetrical. The line whi…

3. ### 6th Grade Math Practice | Table of Contents | Worksheets |Videos |Math

Aug 10, 24 01:59 AM

In 6th grade math practice you will get all types of examples on different topics along with the step-by-step explanation of the solutions.

4. ### 6th Grade Algebra Worksheet | Pre-Algebra worksheets with Free Answers

Aug 10, 24 01:57 AM

In 6th Grade Algebra Worksheet you will get different types of questions on basic concept of algebra, questions on number pattern, dot pattern, number sequence pattern, pattern from matchsticks, conce…