If the line y = m x + 7 3 is normal to the hyperbola x 2 24 - y 2 18 = 1 , then a value of m is:

Step-by-step solution

The given equation of the hyperbola is x 2 24 - y 2 18 = 1 , which is of the form x 2 a 2 - y 2 b 2 = 1 . ⇒ a 2 = 24 ;   b 2 = 18 We know that the equation of a normal of slope m to the hyperbola x 2 a 2 - y 2 b 2 = 1 is y = m x ± m a 2 + b 2 a 2 - b 2 m 2 Thus, the equation of the normal to the given hyperbola is y = m x ± m 24 + 18 24 - 18 m 2 y = m x ± m 42 24 - 18 m 2 Given, the normal is y = m x + 7 3 Hence, comparing the two equations, we get 42 m 24 - 18 m 2 = 7 3 ⇒   42