the foci are ( , ± ) and that of the vertices are ( , ± ). Also, The eccentricity c e . The latus rectum b Example Find the equation of the hyperbola with foci ( , ± ) and vertices ( , ± ). Solution Since the foci is on y-axis, the equation of the hyperbola is of the form – b Since vertices are ( , ± ), a = Also, since foci are ( , ± ); c = and b = c – a = .
Therefore, the equation of the hyperbola is – = , i.e., y – x = . Example Find the equation of the hyperbola where foci are ( , ± ) and the length of the latus rectum is . Solution Since foci are ( , ± ), it follows that c = . Length of the latus rectum = b or b = a Therefore c = a + b ; gives = a + a i.e., a + a – = , So a = – , .
Since a cannot be negative, we take a = and so b = . Therefore, the equation of the required hyperbola is – = , i.e., y – x = MATHEMATICS Fig . EXERCISE . In each of the Exercises to , find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.
x – y = . y – x = . y – x = . In each of the Exercises to , find the equations of the hyperbola satisfying the given conditions.
. Vertices ( ± , ), foci (