axis, the minor axis, the eccentricity and the latus rectum of the ellipse Solution Since denominator of x is larger than the denominator of y , the major Fig . CONIC SECTIONS axis is along the x -axis. Comparing the given equation with b = , we get a = and b = . Also c a – b – Therefore, the coordinates of the foci are (– , ) and ( , ), vertices are (– , ) and ( , ).
Length of the major axis is units length of the minor axis b is units and the eccentricity is and latus rectum is b a = Example Find the coordinates of the foci, the vertices, the lengths of major and minor axes and the eccentricity of the ellipse x + y = . Solution The given equation of the ellipse can be written in standard form as Since the denominator of y is larger than the denominator of x , the major axis is along the y -axis. Comparing the given equation with the standard equation b = , we have b = and a = . Also c = a – b = – and c e Hence the foci are ( , ) and ( , – ), vertices are ( , ) and ( , – ), length of the major axis is units, the length of the minor axis is units and the eccentricity of the ellipse is .
Example Find the equation of the ellipse whose vertices are ( ± , ) and foci are ( ± , ). Solution Since the vertices are on x -axis, the equation will be of the form b = , where a is the semi-major axis. MATHEMATICS Given that a = , c = ± . Therefore, from the relation c = a – b , we get = – b , i.e., b = Hence the