Equation of ellipse





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

Parents and teachers can guide the students to go through the examples discussed in 'Equation of ellipse' step by step, and ask them to solve the practice problems on their own. Solutions are given for the practice problems. Students can follow the steps and master in these type of problems. If you have any doubt please contact us, through mail we will help you to clear your doubts.





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