Through the point (4, 5), a straight line is drawn making positive intercepts on the coordinate axes. The area of the triangle thus formed is least, when the ratio of the intercepts on the X and Y axes is

Step-by-step solution

Since, intercept form of line is x a + y b =1 where X-intercept = a, Y-intercept = b Hence, equation of line AB x m + y n =1 Line (i) passes through (4,5) . aligned 4 m & + 5 n =1 4 m & =1- 5 n m & = 4 n n-5 aligned gathered Area of O A B= 1 2 O A O B A= 1 2 m n= 1 2 ( 4 n n-5 ) n A= 2 n^2 n-5 d A d n =2 (n-5) 2 n-n^2(1) (n-5)^2 d A d n =2 2 n^2-10 n-n^2 (n-5)^2 = 2 n^2-20 n (n-5)^2 gathered Area is least for some value of n . array ll & d A d n =0 & 2 n^2-20 n=0 & 2 n(n-10)=0 n=0,10 At & n=10, m= 4 n n-5 = 4 10 10