Find \(m\) if the points \((5,1),\ (-2,-3),\ (8,2m)\) are collinear.
\(m=\dfrac{19}{14}\)
Step 1: Three points are collinear if the area of the triangle formed by them is 0.
Step 2: Use the area formula for points \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\):
\(\text{Area} = \dfrac{1}{2}\left|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\right|\).
Step 3: Substitute \((5,1)\), \((-2,-3)\), \((8,2m)\).
Step 4: Compute the expression inside the absolute value:
\(5\big((-3)-2m\big) + (-2)\big(2m-1\big) + 8\big(1-(-3)\big)\)
Simplify term by term:
Add them:
\((-15 - 10m) + (-4m + 2) + 32 = 19 - 14m\)
Step 5: For collinearity, area must be zero, so the expression must be zero:
\(|19 - 14m| = 0 \Rightarrow 19 - 14m = 0\)
Step 6: Solve for \(m\).
\(14m = 19\)
\(m = \dfrac{19}{14}\)
Final Answer: \(m = \dfrac{19}{14}\).