If the points A(1, 2), O(0, 0) and C(a, b) are collinear, then ______.

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If the points A(1, 2), O(0, 0) and C(a, b) are collinear, then 2a = b. Explanation: Let the given points are A = (x1, y1) = (1, 2), B = (x2, y2) = (0, 0) and C = (x3, y3) = (a, b). ∵ Area of ΔABC Δ = `$\frac{1}{2}$[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)] $\therefore$ Δ = $\frac{1}{2}$[1(0 - b) + 0(b - 2) + a( 2 - 0)] = $\frac{1}{2}$(-b + 0 + 2a) = $\frac{1}{2}$(2a - b)` Since, the points A(1, 2), B(0, 0) and C(a, b) are collinear, then area of ΔABC should be equal to zero. i.e., Area of ΔABC = 0 $\Rightarrow$ $\frac{1}{2}$(2a - b) = 0 $\Rightarrow$ 2a – b = 0 $\Rightarrow$ 2a = b Hence, the required relation is 2a = b.

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