In this page 'Equation of ellipse' we are going to examples which describes how to get the equation of the ellipse from the given foci, eccentricity and directrix.

**Example 1:**

Find the equation of the ellipse whose focus is (-1,1), eccentricity is 1/2 and whose directrix is x-y+3=0.

**Solution:**

The focus is (-1,1).

Directrix is x-y+3=0 and e =1/2.

Let P(x₁,y₁) be any point on the ellipse, then SP² = e²PM² where PM is perpendicular distance from x-y+3=0

(x₁+1)² + (y₁-1)² = 1/4 [(x₁-y₁+3)/√(1+1)]²

8(x₁+1)² +8 (y₁-1)² = (x₁-y₁+3)²

7x₁²+2x₁y₁+7y₁²+10x₁-10y₁+7=0

Locus of (x₁,y₁) i.e., the equation of the required ellipse is

** 7x²+2xy+7y+10x-10y+7=0**

**Example 2:**

Find the equation of the ellipse whose foci are (2,0) and (-2,0) and eccentricity is 1/2.

**Solution:**

S(ae,0) and S'(ae,0) are the foci of the ellipse (x²/a²)+(y²/b²) =1.

The foci are (2,0) and (-2,0) and e=1/2

ae = 2 and e = 1/2

a= 4 and a² =16

The center C is the mid point of SS' is (0,0). S and S' are on the x axis. So the equation of the ellipses of the form (x²/a²)+(y²/b²) =1

b² = a²(1-e²) = 16(1-1/4) = 12.

The equation of the required ellipse is **(x²/16)+(y²/12) =1.**

We know that the equation of the ellipse is (x²/a²)+(y²/b²) =1, where a is the major axis (which is horizontal X axis), b is the minor axis and a>b here.

If the equation is ,(x²/b²)+(y²/a²) =1 then here a is the major axis which is vertical Y axis, b is the minor axis and a>b.

**Example :**

In the equation (x²/16)+(y²/9) =1 we have a =4 and b=3. a>b, so the **major axis is X axis**.

In the equation (x²/16)+(y²/25) =1, we have a =4 and b=5 and >a.

So the** major axis is** the vertical axis that is **Y axis**

__Practice problems:__

1. Find the equation of the ellipse whose focus is (1,2), directrix is 2x-3y+6=0 and the eccentricity is 2/3. Solution

2. Find the equation of the ellipse whose foci are (4,0) and (-4,0) and e =1/3. Solution

**Related Topics**

**Equation of Parabola****Equation of ellipse****Examples of ellipse****Worksheet of parabola****More on analytical geometry****Analytical geometry worksheets**

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