Using determinants show that the following points are collinear:
(i) (5, 5), (−5, 1) and (10, 7)
(ii) (1, −1), (2, 1) and (4, 5)
(iii) (3, −2), (8, 8) and (5, 2)
(iv) (2, 3), (−1, −2) and (5, 8)

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Solution

(i) If the points (5, 5), (−5, 1) and (10, 7) are collinear, then

$∆ = |\begin{matrix} 5 & 5 & 1 \\ - 5 & 1 & 1 \\ 10 & 7 & 1 \end{matrix}| = 0 = |\begin{matrix} 5 & 5 & 1 \\ - 10 & - 4 & 0 \\ 10 & 7 & 1 \end{matrix}| [Applying R_{2} \to R_{2} - R_{1}] = |\begin{matrix} 5 & 5 & 1 \\ - 10 & - 4 & 0 \\ 5 & 2 & 0 \end{matrix}| [Applyin g R_{3} \to R_{3} - R_{1}] = |\begin{matrix} - 10 & - 4 \\ 5 & 2 \end{matrix}| = - 20 + 20 = 0$

Thus, these points are colinear.

(ii) If the points (1, −1), (2, 1) and (4, 5) are collinear, then

$∆ = |\begin{matrix} 1 & - 1 & 1 \\ 2 & 1 & 1 \\ 4 & 5 & 1 \end{matrix}| = 0 = |\begin{matrix} 1 & - 1 & 1 \\ 1 & 2 & 0 \\ 4 & 5 & 1 \end{matrix}| [Applying R_{2} \to R_{2} - R_{1}] = |\begin{matrix} 1 & - 1 & 1 \\ 1 & 2 & 0 \\ 3 & 6 & 0 \end{matrix}| [Applying R_{3} \to R_{3} - R_{1}] = |\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}| = 6 - 6 = 0$

Thus, these points are collinear.

(iii) If the points (3, −2), (8, 8) and (5, 2) are collinear, then

$∆ = |\begin{matrix} 3 & - 2 & 1 \\ 8 & 8 & 1 \\ 5 & 2 & 1 \end{matrix}| = 0 = |\begin{matrix} 3 & - 2 & 1 \\ 5 & 10 & 0 \\ 5 & 2 & 1 \end{matrix}| [Applying R_{2} \to R_{2} - R_{1}] = |\begin{matrix} 3 & - 2 & 1 \\ 5 & 10 & 0 \\ 2 & 4 & 0 \end{matrix}| [Applying R_{3} \to R_{3} - R_{1}] = |\begin{matrix} 5 & 10 \\ 2 & 4 \end{matrix}| = 20 - 20 = 0$

Thus the points are colinear.

(iv) If the points (2, 3), (−1, −2) and (5, 8) are collinear, then

$∆ = |\begin{matrix} 2 & 3 & 1 \\ - 1 & - 2 & 1 \\ 5 & 8 & 1 \end{matrix}| = 0 = |\begin{matrix} 2 & 3 & 1 \\ - 3 & - 5 & 0 \\ 5 & 8 & 1 \end{matrix}| [Applying R_{2} \to R_{2} - R_{1}] = |\begin{matrix} 2 & 3 & 1 \\ - 3 & - 5 & 0 \\ 3 & 5 & 0 \end{matrix}| [Applying R_{3} \to R_{3} - R_{1}] = |\begin{matrix} - 3 & - 5 \\ 3 & 5 \end{matrix}| = - 15 + 15 = 0$

Thus the points are colinear.

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Similar questions

Using the distance formula, show that the given points are collinear:

(i) (1, -1), (5, 2) and 9, 5 (ii) (6, 9), (0, 1) and (-6, -7)

(iii) (-1, -1), (2, 3) and (8, 11) (iv) (-2, 5), (0, 1) and (2, 3).

Show that the following points are collinear :

(i) A (2, -2), B(-3, 8) and C(-1, 4)

(ii) A(-5, 1), B(5,5) and C(10, 7)

(iii) A(5, 1), B(1, -1) and C(11, 4)

(iv) A(8, 1), B(3, -4) and C(2, -5)

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