a c c which on simplification gives c c ) Squaring again and simplifying, we get c = i.e., b = (Since c = a – b ) Hence any point on the ellipse satisfies b = . ... ( ) Conversely, let P ( x , y ) satisfy the equation ( ) with < c < a . Then y = b Therefore, PF c ) b c ) a c c (since b = a – c ) cx c = + Similarly PF = c CONIC SECTIONS Hence PF + PF = c c a – ...
( ) So, any point that satisfies b = , satisfies the geometric condition and so P( x, y) lies on the ellipse. Hence from ( ) and ( ), we proved that the equation of an ellipse with centre of the origin and major axis along the x -axis is b = . Discussion From the equation of the ellipse obtained above, it follows that for every point P ( x , y ) on the ellipse, we have b ≤ , i.e., x ≤ a , so – a ≤ x ≤ a. Therefore, the ellipse lies between the lines x = – a and x = a and touches these lines.
Similarly, the ellipse lies between the lines y = – b and y = b and touches these lines. Similarly, we can derive the equation of the ellipse in Fig . (b) as b = . These two equations are known as standard equations of the ellipses.