A Note The standard equations of hyperbolas have transverse and conjugate axes as the coordinate axes and the centre at the origin. However, there are hyperbolas with any two perpendicular lines as transverse and conjugate axes, but the study of such cases will be dealt in higher classes. From the standard equations of hyperbolas (Fig11. ), we have the following observations: .
Hyperbola is symmetric with respect to both the axes, since if ( x , y ) is a point on the hyperbola, then (– x , y ), ( x , – y ) and (– x , – y ) are also points on the hyperbola. MATHEMATICS . The foci are always on the transverse axis. It is the positive term whose denominator gives the transverse axis.
For example, – has transverse axis along x -axis of length , while – has transverse axis along y-axis of length . . . Latus rectum Definition Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.
As in ellipse, it is easy to show that the length of the latus rectum in hyperbola is b a . Example Find the coordinates of the foci and the vertices, the eccentricity,the length of the latus rectum of the hyperbolas: – = , (ii) y – x = Solution (i) Comparing the equation – = with the standard equation – b Here, a = , b = and c = b Therefore, the coordinates of the foci are ( ± , ) and that of vertices are ( ± , ).Also, The eccentricity e = c a = . The latus rectum b (ii) Dividing the equation by on both sides, we have – Comparing the equation with the standard equation – b = , we find that a = , b = and c b CONIC SECTIONS Therefore, the coordinates of