(Since c – a = b ) Hence any point on the hyperbola satisfies – b = . Conversely, let P( x , y ) satisfy the above equation with < a < c . Then y = b x – a Therefore, PF = + c = + x – a c b = a + c Similarly, PF = a – a c x In hyperbola c > a ; and since P is to the right of the line x = a , x > a , c a x > a . Therefore, a – c a x becomes negative.
Thus, PF = c a x – a. CONIC SECTIONS Therefore PF – PF = a + c a x – cx a + a = a Also, note that if P is to the left of the line x = – a , then PF c – a , PF = a – c x In that case P F – PF = a . So, any point that satisfies – b = , lies on the hyperbola. Thus, we proved that the equation of hyperbola with origin ( , ) and transverse axis along x -axis is – b = .
A Note A hyperbola in which a = b is called an equilateral hyperbola . Discussion From the equation of the hyperbola we have obtained, it follows that, we have for every point ( x , y ) on the hyperbola, b = + ≥ . i.e, a x ≥ , i.e., x ≤ – a or x ≥ a . Therefore, no portion of the curve lies between the lines x = + a and x = – a, (i.e.
no real intercept on the conjugate axis). Similarly, we can derive the equation of the hyperbola in Fig . (b) as b = These two equations are known as the standard equations of hyperbolas .