Show that points A (a,b+c),B (b,c+a),C (c,a+b) are collinear
Area of ∆ ABC is given by the relation,
△=\(\frac{1}{2}\)\(\begin{vmatrix}a&b+c&1\\b&c+a&1\\c&a+b&1\end{vmatrix}\)
=\(\frac{1}{2}\)\(\begin{vmatrix}a&b+c&1\\b-a&a-b&0\\c-a&a-c&0\end{vmatrix}\) (Applying R2\(\to\) R2-R1 and R33\(\to\)R3-R1)
=\(\frac{1}{2}\)(a-b)(c-a)\(\begin{vmatrix}a&b+c&1\\-1&1&0\\1&-1&0\end{vmatrix}\)
=\(\frac{1}{2}\)(a-b)(c-a)\(\begin{vmatrix}a&b+c&1\\-1&1&0\\1&-1&0\end{vmatrix}\) (applying R3\(\to\) R3+R2)
=0 (All elements of R3 are 0)
Thus, the area of the triangle formed by points A, B, and C is zero
Hence, the points A, B, and C are collinear.
A settling chamber is used for the removal of discrete particulate matter from air with the following conditions. Horizontal velocity of air = 0.2 m/s; Temperature of air stream = 77°C; Specific gravity of particle to be removed = 2.65; Chamber length = 12 m; Chamber height = 2 m; Viscosity of air at 77°C = 2.1 × 10\(^{-5}\) kg/m·s; Acceleration due to gravity (g) = 9.81 m/s²; Density of air at 77°C = 1.0 kg/m³; Assume the density of water as 1000 kg/m³ and Laminar condition exists in the chamber.
The minimum size of particle that will be removed with 100% efficiency in the settling chamber (in $\mu$m is .......... (round off to one decimal place).