1) If the points (a,0) (0,b) and (1,1) are collinear then show that 1/a+1/b=1 - Maths - Coordinate Geometry

Question

1) If the points (a,0) (0,b) and (1,1) are collinear then show that
1/a+1/b=1

Lalit Mehra · Accepted Answer

Area of triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) $is \frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]$ Suppose the given point be A(a, 0), B(0, b) and C(1, 1) The given points are collinear âˆ´ Area of Î”ABC = 0 $\Rightarrow \frac{1}{2} [a (b - 1) + 0 (1 - 0) + 1 (0 - b)] = 0 \Rightarrow a b - a - b = 0 \Rightarrow a + b = a b \Rightarrow \frac{a}{a b} + \frac{b}{a b} = 1 \Rightarrow \frac{1}{a} + \frac{1}{b} = 1$ Thus, the given points are collinear if $\frac{1}{a} + \frac{1}{b} = 1$