​​​If (3,2) (-3,5) and (x,y) are collinear then show that x+2y-7=0.

Dear student
$$\begin{gathered} \mathrm{Given} :\mathrm{A}\left(3,2\right) , \mathrm{B}\left(-3,5\right) \mathrm{and} \mathrm{C}\left(\mathrm{x},\mathrm{y}\right) \mathrm{are} \mathrm{collinear} \\ \mathrm{Then} \mathrm{area} \mathrm{of} \mathrm{the} \mathrm{triangle} \mathrm{formed} \mathrm{by} \mathrm{the} \mathrm{points} \mathrm{A}\left(3,2\right) , \mathrm{B}\left(-3,5\right) \mathrm{and} \mathrm{C}\left(\mathrm{x},\mathrm{y}\right) \\ =\mathrm{zero} \\ \Rightarrow \frac{1}{2}\left[{\mathrm{x}}_1\left({\mathrm{y}}_2-{\mathrm{y}}_3\right)+{\mathrm{x}}_2\left({\mathrm{y}}_3-{\mathrm{y}}_1\right)+{\mathrm{x}}_3\left({\mathrm{y}}_1-{\mathrm{y}}_2\right)\right]=0 \\ \Rightarrow {\mathrm{x}}_1\left({\mathrm{y}}_2-{\mathrm{y}}_3\right)+{\mathrm{x}}_2\left({\mathrm{y}}_3-{\mathrm{y}}_1\right)+{\mathrm{x}}_3\left({\mathrm{y}}_1-{\mathrm{y}}_2\right)=0 \\ \Rightarrow 3\left(5-\mathrm{y}\right)+\left(-3\right)\left(\mathrm{y}-2\right)+\mathrm{x}\left(2-5\right)=0 \\ \Rightarrow 15-3\mathrm{y}-3\mathrm{y}+6+2\mathrm{x}-5\mathrm{x}=0 \\ \Rightarrow -3\mathrm{x}-6\mathrm{y}+21=0 \\ \Rightarrow \mathrm{x}+2\mathrm{y}-7=0 \\ \mathrm{Hence} \mathrm{proved} \end{gathered}$$
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