Question

find the equation of an ellipse whose foci are 2, 3 and minus 23 and whose semi minor axis is root 5

Answer 1

Step-by-step explanation:

Given  

find the equation of an ellipse whose foci are (2,3) and (-2,3) and whose semi minor axis is root 5

We know that (x1,y1)(x2,y2) will be x1 + x2 / 2 , y1 + y2 / 2

                                                         = 2 + (-2) / 2 , 3 + 3 / 2

                                                        = (0,3 )

So centre of ellipse will be (0,3)

We have x = +-ae and y = 0 + 3 = 3

So we get (+-ae, 3)

 So we get (+-2,3) (since ae = 2)

We know that e = √1 – b^2 / a^2

           So 2 = a√1 – b^2/a^2

Squaring both sides we get and b = √5

          4 = (1 - √5^2 / a^2 )  

         4 = (a^2 – 5 / a^2)  

  So a^2 = 9

Or a = 3 and b = √5

We know that general equation is (x – h)^2 / a^2 + (y – b)^2 / b^2 = 1

                                           = (x – 0)^2 / 9 + (y – 3)^2 / 5 = 1

                                          = x^2 / 9 + (y – 3)^2 / 5 = 1

This is the required equation of the ellipse.