A triangle has two fixed vertices A (a, 0) and B (0, b) . Let its third vertex C is moving along the line x = y . If s is the area of triangle ABC, then ds dx =

Step-by-step solution

Let the coordinates of the vertices of the triangle be A(a, 0) , B(0, b) , and C(x, x) since C lies on the line y = x . The area of the triangle s is given by the determinant formula: s = 1 2 vmatrix a & 0 & 1 0 & b & 1 x & x & 1 vmatrix Expanding the determinant along the first row: s = 1 2 [a(b - x) - 0 + 1(0 - bx)] s = 1 2 (ab - ax - bx) s = 1 2 [ab - (a + b)x] Differentiating s with respect to x : ds dx = d dx [ 1 2 (ab - (a + b)x) ] ds dx = - ( a + b 2 ) Answer: - ( a + b 2 )