Examples on Ellipse Formula

Example 1: Find the equation of ellipse if the endpoints of the major axis lie on (-10,0) and (10,0) and endpoints of the minor axis lie on (0,-5) and (0,5). 

Solution: 

Since the major axis is x-axis, the ellipse equation should be, 

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

2a = 20 

⇒a = 10 

2b = 10 

⇒b = 5

\frac{x^2}{10^2} + \frac{y^2}{5^2} = 1

Example 2: Find the equation of an ellipse with origin as centre and x-axis as major axis. Given that the distance between two foci is 10cm, e = 0.4 and b = 4cm

Solution: 

Standard equation of the ellipse is, 

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

We know b = 4, e = 0.4 and c = 10. 

c = ae \\ 10 = a(0.4) \\ 25 = a

Thus, now we have a = 25 and b = 4 

So, the equation of ellipse is, 

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \\ \frac{x^2}{25^2} + \frac{y^2}{4^2} = 1

Example 3: Find the equation of an ellipse whose major axis is 40cm and foci lie on (5,0) and (-5,0). 

Solution: 

\frac{x^2}{a^2} + \frac{x^2}{b^2} = 1

a = \frac{40}{2} = 20

We know c = 10

 c² = a² – b² 

10² = 20² – b²

 b² = 20² – 10²

b² = 300

Thus, the equation becomes, 

\frac{x^2}{400} + \frac{x^2}{300} = 1

Example 4: Find the equation of an ellipse whose major axis is 40cm and foci lie on (0,5) and (0,-5). 

Solution: 

Since the foci lie on y-axis. The major axis is on y-axis. Thus, the ellipse is of the form, 

\frac{x^2}{b^2} + \frac{x^2}{a^2} = 1

a = \frac{40}{2} = 20

We know c = 10

 c² = a² – b² 

10² = 20² – b²

 b² = 20² – 10²

b² = 300

Thus, the equation becomes, 

\frac{x^2}{300} + \frac{x^2}{400} = 1

Example 5: Find the equation of ellipse if the major axis is the x-axis and the minor axis is the y-axis and (4,3) and (-1,4) lie on the ellipse. 

Solution: 

Standard equation of the ellipse is, 

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

These points must satisfy the equation. (4,3) and (-1, 4). 

\frac{4^2}{a^2} + \frac{(-3)^2}{b^2} = 1 \\ \frac{16}{a^2} + \frac{9}{b^2} = 1 \\ 

\frac{(-1)^2}{a^2} + \frac{4^2}{b^2} = 1 \\ \frac{1}{a^2} + \frac{16}{b^2} = 1 \\ 

Let’s say, x = \frac{1}{a} \text{ and } y = \frac{1}{b}

16x + 9b = 1 

x + 16b = 1

Solving the equations, 

We find that a = \frac{247}{7} \text{ and } b = \frac{247}{15}

An ellipse is a set of points such that the sum of the distances from any point on the ellipse to two fixed points (foci) is constant. In this article, we will learn about, Ellipse definition, Ellipse formulas and others in detail.