BF So that, BF – BF = BA + AF – BF = BA = a Fig . Fig . CONIC SECTIONS . .
Eccentricity Definition Just like an ellipse, the ratio e = c a is called the eccentricity of the hyperbola . Since c ≥ a , the eccentricity is never less than one. In terms of the eccentricity, the foci are at a distance of ae from the centre. .
. Standard equation of Hyperbola The equation of a hyperbola is simplest if the centre of the hyperbola is at the origin and the foci are on the x -axis or y -axis. The two such possible orientations are shown in Fig11. .
We will derive the equation for the hyperbola shown 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 through O through F be the positive x -axis and that through F as the negative x -axis.
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 hyperbola such that the difference of the distances from P to the farther point minus the closer point be a.
So given , PF – PF = a Fig . (a) (b) Fig . MATHEMATICS Using the distance formula, we have c – x – c i.e., c x – c Squaring both side, we get ( x + c ) + y = a + a x – c + ( x – c ) + y and on simplifying, we get cx – a = x – c On squaring again and further simplifying, we get – c – a i.e., – b