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प्रश्न
Determine whether the points are collinear.
L(–2, 3), M(1, –3), N(5, 4)
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उत्तर
L(–2, 3), M(1, –3), N(5, 4)
According to distance formula,
d(L, M) = `sqrt((x_2 – x_1)^2 + (y_2 – y_1)^2)`
d(L, M) = `sqrt([1 – (–2)]^2 + (–3 – 3)^2)`
d(L, M) = `sqrt((1 + 2)^2 + (–3 – 3)^2)`
d(L, M) = `sqrt((3)^2 + (–6)^2)`
d(L, M) = `sqrt(9 + 36)`
d(L, M) = `sqrt(45)`
d(L, M) = `sqrt(9 × 5)`
∴ d(L, M) = `3sqrt(5)` ...(1)
d(M, N) = `sqrt((x_2 – x_1)^2 + (y_2 – y_1)^2)`
d(M, N) = `sqrt((5 – 1)^2 + [4 – (– 3)]^2)`
d(M, N) = `sqrt((5 – 1)^2 + (4 + 3)^2)`
d(M, N) = `sqrt((4)^2 + (7)^2)`
d(M, N) = `sqrt(16 + 49)`
∴ d(M, N) = `sqrt(65)` ...(2)
d(L, N) = `sqrt((x_2 – x_1)^2 + (y_2 – y_1)^2)`
d(L, N) = `sqrt([5 – (– 2)]^2 + (4 – 3)^2)`
d(L, N) = `sqrt((5 + 2)^2 + (4 – 3)^2)`
d(L, N) = `sqrt((7)^2 + (1)^2)`
d(L, N) = `sqrt(49 + 1)`
d(L, N) = `sqrt(50)`
d(L, N) = `sqrt(25 × 2)`
∴ d(L, N) = `5sqrt(2)` ...(3)
From (1), (2), and (3),
Sum of two sides is not equal to the third side.
Hence, the given points are not collinear.
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Case Study -2
A hockey field is the playing surface for the game of hockey. Historically, the game was played on natural turf (grass) but nowadays it is predominantly played on an artificial turf.
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