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  • Find the value of m if the points (5, 1), (–2, –3) and (8, 2m ) are collinear.

Find the value of m if the points (5, 1), (–2, –3) and (8, 2m ) are collinear.

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Solution. 

Area \, of \, triangle = \frac{1}{2}\left [ x_{1} \left ( y_{2} -y_{3}\right )+x_{2} \left ( y_{3} -y_{1}\right )+x_{3} \left ( y_{1} -y_{2}\right )\right ]
Given points are A(5, 1), B(–2, –3), C(8, 2m)
If points are collinear then the area of triangle = 0
= \frac{1}{2}\left [ x_{1} \left ( y_{2} -y_{3}\right )+x_{2} \left ( y_{3} -y_{1}\right )+x_{3} \left ( y_{1} -y_{2}\right )\right ]= 0
\Rightarrow \left [ x_{1} \left ( y_{2} -y_{3}\right )+x_{2} \left ( y_{3} -y_{1}\right )+x_{3} \left ( y_{1} -y_{2}\right )\right ]= 0
\Rightarrow 5(–3 – 2m) – 2(2m – 1) + 8(1 + 3) = 0
\Rightarrow – 15 – 10m – 4m + 2 + 32 = 0
\Rightarrow – 14m + 19 = 0
\Rightarrow 14m = 19
\Rightarrow m= \frac{19}{14}

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