A straight line is drawn through a given point P(1,4) . The least value of the sum of the intercepts on the co-ordinate axes is

Step-by-step solution

Let the equation of the line passing through P(1, 4) be x a + y b = 1 , where a and b are the x-intercept and y-intercept respectively. Since the line passes through (1, 4) , we have 1 a + 4 b = 1 . We want to minimize the sum of intercepts S = a + b . From the equation 1 a + 4 b = 1 , we can express b in terms of a : 4 b = 1 - 1 a = a-1 a b = 4a a-1 . The sum S = a + 4a a-1 . To minimize S , we differentiate with respect to a : dS da = 1 + (a-1)4 - 4a(1) (a-1)^2 = 1 - 4 (a-1)^2 . Setting dS da = 0 , we get (a-1)^2