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Question
If the points A(1, 2), O(0, 0) and C(a, b) are collinear, then ______.
Options
a = b
a = 2b
2a = b
a = – b
Solution
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 = `1/2[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)]`
∴ Δ = `1/2[1(0 - b) + 0(b - 2) + a( 2 - 0)]`
= `1/2(-b + 0 + 2a)`
= `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
⇒ `1/2(2a - b)` = 0
⇒ 2a – b = 0
⇒ 2a = b
Hence, the required relation is 2a = b.
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