units. Example . Find the equation of the ellipse with foci ( , ) ± , vertices ( , ) ± . From Fig.
. , we get SS ′ = c and c = ; ′ A A ⇒ c = and a = , ⇒ b = a Major axis is along x -axis, since a > Centre is ( , ) and Foci are ( , ) ± . Therefore, equation of the ellipse is x Example . Find the equation of the ellipse whose eccentricity is , one of the foci is( , ) and a directrix is x = .
Also find the length of the major and minor axes of the ellipse. By the definition of a conic, SP PM = e or SP e PM Then, ) + = x − ⇒ + = ⇒ + ) = + × − ⇒ = which is in the standard form. Therefore, the length of major axis = a = the length of minor axis = b = Fig. .
C S ' S ( , ) A' (– , ) A ( , ) (– , ) ( , ) - - Example . Find the foci, vertices and length of major and minor axis of the conic Completing the square on x and y of = , gives = − ) + ) = . Dividing both sides by , the equation reduces to ) + = . This is an ellipse with centre ( , ) − , major axis is parallel to x -axis, length of major axis is and length of minor axis is .
Vertices are ( , ) and ( , ) − . Now, c =