(ii) When c = a , then b = . The ellipse reduces to the line segment F F joining the two foci (Fig11. ). .
. Eccentricity Definition The eccentricity of an ellipse is the ratio of the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse (eccentricity is denoted by e ) i.e., c e Fig . Fig . Fig .
CONIC SECTIONS Then since the focus is at a distance of c from the centre, in terms of the eccentricity the focus is at a distance of ae from the centre. . . Standard equations of an ellipse The equation of an ellipse is simplest if the centre of the ellipse is at the origin and the foci are Fig .
(a) on the x -axis or y -axis. The two such possible orientations are shown in Fig . . We will derive the equation for the ellipse shown above in Fig .
(a) with foci on the x -axis. Let F and F be the foci and O be the mid- point of the line segment F F . Let O be the origin and the line from O through F be the positive x -axis and that through F as the negative x -axis. Let, the line through O perpendicular to the x -axis be the y -axis.
Let the coordinates of F be (– c , ) and F be ( c , ) (Fig . ). Let P( x , y ) be any point on the ellipse such that the sum of the distances from P to the two foci be a so given PF + PF = a . ...
( ) Using the distance formula, we have c c = a i.e., ) c = a – ) c Fig . b MATHEMATICS Squaring both sides, we get ( x + c ) + y = a –