A triangle ABC is formed by lines 2x-3y-6=0; 3x-y+3=0 and 3x+4y-12=0. If the points P(a,0) and Q(0,b) always lie on or inside the triangle ABC then find the range of a and b both .

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$$\begin{gathered} \mathrm{We} \mathrm{draw} \mathrm{the} \mathrm{triangle} \mathrm{on} \mathrm{coordinate} \mathrm{axes} \mathrm{as} \mathrm{shown}, \\ \mathrm{P}\left(\mathrm{a},0\right) \\ \mathrm{Q}\left(0,\mathrm{b}\right) \\ \mathrm{for} \mathrm{them} \mathrm{to} \mathrm{lie} \mathrm{on} \mathrm{the} \mathrm{triangle} \mathrm{they} \mathrm{must} \mathrm{satisfy} \mathrm{any} \mathrm{of} \mathrm{the} \mathrm{three} \mathrm{given} \mathrm{equations} \mathrm{of} \mathrm{sides}: \\ 2\mathrm{x}-3\mathrm{y}-6=0 \\ 2\mathrm{a}=6 \\ \mathrm{a}=3 \\ -3\mathrm{b}=6 \\ \mathrm{b}=-2 \\ 3\mathrm{x}-\mathrm{y}+3=0 \\ 3\mathrm{a}=-3 \\ \mathrm{a}=-1 \\ \mathrm{or} \\ -\mathrm{b}+3=0 \\ \mathrm{b}=3 \\ 3\mathrm{x}+4\mathrm{y}-12=0 \\ 3\mathrm{a}=12 \\ \mathrm{a}=4 \\ 4\mathrm{b}=12 \\ \mathrm{b}=3 \\ \mathbf{so}\mathbf{ }\mathbf{if}\mathbf{ }\mathbf{a}\mathbf{\in }\left\{\mathbf{3}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{4}\right\}\mathbf{,}\mathbf{ }\mathbf{b}\mathbf{\in }\left\{\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{3}\right\}\mathbf{ }\mathbf{then}\mathbf{ }\mathbf{P}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{Q}\mathbf{ }\mathbf{lie}\mathbf{ }\mathbf{on}\mathbf{ }\mathbf{triangle} \\ \mathrm{From} \mathrm{figure} \mathrm{it} \mathrm{is} \mathrm{clear} \mathrm{origin}\left(0,0\right) \mathrm{is} \mathrm{internal} \mathrm{point} \mathrm{of} \mathrm{triangle} \\ 2\mathrm{x}-3\mathrm{y}-6=0 \\ \mathrm{put} \mathrm{x}=0,\mathrm{y}=0 \\ -6<0 \\ \mathrm{hence} \left(\mathrm{a},0\right) \mathrm{and} \left(0,\mathrm{b}\right) \mathrm{should} \mathrm{also} \mathrm{be} -\mathrm{ve} \mathrm{when} \mathrm{put} \\ 2\mathrm{a}-6<0 \\ 2\mathrm{a}<6 \\ \mathrm{a}<3 \\ -3\mathrm{b}-6<0 \\ -3\mathrm{b}<6 \\ \mathrm{b}<-2 \\ 3\mathrm{x}-\mathrm{y}+3=0 \\ \mathrm{put} \mathrm{x}=0,\mathrm{y}=0 \\ 3>0 \\ \mathrm{hence} \\ 3\mathrm{a}+3>0 \\ \mathrm{a}>-1 \\ -\mathrm{b}+3>0 \\ \mathrm{b}<3 \\ 3\mathrm{x}+4\mathrm{y}-12=0 \\ \mathrm{put} \mathrm{x}=0,\mathrm{y}=0 \\ -12<0 \\ \mathrm{hence}, \\ 3\mathrm{a}-12<0 \\ 3\mathrm{a}<12 \\ \mathrm{a}<4 \\ 4\mathrm{b}-12<0 \\ 4\mathrm{b}<12 \\ \mathrm{b}<3 \\ \mathrm{so} \mathrm{we} \mathrm{have} \\ \mathrm{a}<3 \\ \mathrm{a}<4 \\ \mathrm{a}>-1 \\ \mathrm{and} \\ \mathrm{b}<3 \\ \mathrm{b}<-2 \\ \mathbf{taking}\mathbf{ }\mathbf{intersection}\mathbf{ }\mathbf{we}\mathbf{ }\mathbf{get}\mathbf{,} \\ \mathbf{a}\mathbf{\in }\left(\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{3}\right) \\ \mathbf{b}\mathbf{\in }\left(\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\right) \\ \mathbf{for}\mathbf{ }\mathbf{P}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{Q}\mathbf{ }\mathbf{to}\mathbf{ }\mathbf{be}\mathbf{ }\mathbf{inside}\mathbf{ }\mathbf{triangle}\mathbf{.} \end{gathered}$$


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