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Let the given points be A(1, 1), B(3, K) and C(-1, 4). Here, we have x 1 = 1, y 1 = 1 x 2 = 3, y 2 = k and x 3 = -1, y 3 = 4 Given three points, will be collinear if, x 1 (y 2 - y 3 ) + x 2 (y 3 -y 1 ) + x 3 (y 1 - y 2 ) = 0 ⇒ 1(k -4) + 3(4 - 1) + (-1)(1 - k) = 0 ⇒ k - 4 + 3 × 3 + (-1 + k) = 0 ⇒ k - 4 + 9 - 1 + k = 0 ⇒ 2K + 4 = 0 ⇒ 2K = -4 Hence the value of K = -2

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Coordinate Geometry

Short Answer Type

231. Find a relation between x and y if the points (x, y); (1, 2) and (7, 0) are collinear.

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232. Find a relation between x and y if the points (2, 1), (x, y) and (7, 5) are collinear.

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233. The coordinates of the vertices of ∆ABC are A(4, 1), B(-3, 2) and C(0, k). Given that the area of ABC is 12 unit², find the value of k.

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234. Find the area of the ∆ABC with vertices A(-5, 7), B(-4, -5), and C(4, 5).

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235. For what value of k are the points (1, 1), (3, k) and (-1, 4) collinear.

Let the given points be A(1, 1), B(3, K) and C(-1, 4).
Here, we have
x₁ = 1,    y₁ = 1
x₂ = 3,    y₂ = k
and    x₃ = -1, y₃ = 4
Given three points, will be collinear if,
x₁ (y₂ - y₃) + x₂(y₃ -y₁) + x₃ (y₁ - y₂) = 0
⇒    1(k -4) + 3(4 - 1) + (-1)(1 - k) = 0
⇒    k - 4 + 3 × 3 + (-1 + k) = 0
⇒    k - 4 + 9 - 1 + k = 0
⇒    2K + 4 = 0
⇒    2K = -4
Hence the value of    K = -2