if the points(a²,a) and (3,-2) lie on opposite sides of the line x+y+1=0 then belongs to the interval.

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{x}+\mathrm{y}+1=0 \\ \mathrm{Put} \left(\mathrm{x}, \mathrm{y}\right)=\left(3,-2\right) \mathrm{we} \mathrm{get}, \\ 3-2+1=2>0 \\ \mathrm{As} \mathrm{given} \mathrm{points} \mathrm{lie} \mathrm{on} \mathrm{the} \mathrm{opposite} \mathrm{side}, \\ \mathrm{When} \mathrm{we} \mathrm{put} \left({\mathrm{a}}^2,\mathrm{a}\right) \mathrm{in} \mathrm{line} \mathrm{we} \mathrm{have} \\ {\mathrm{a}}^2+\mathrm{a}+1<0 ; \mathrm{condition}\left(\mathrm{i}\right) \\ \mathbf{In}\mathbf{ }\mathbf{any}\mathbf{ }\mathbf{quadratic}\mathbf{ }\mathbf{function}\mathbf{ }\mathbf{f}\left(\mathbf{x}\right)\mathbf{=}{\mathbf{px}}^{\mathbf{2}}\mathbf{+}\mathbf{qx}\mathbf{+}\mathbf{r} \\ \mathbf{if}\mathbf{ }\mathbf{p}\mathbf{>}\mathbf{0}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{D}\mathbf{=}{\mathbf{q}}^{\mathbf{2}}\mathbf{-}\mathbf{4}\mathbf{pr}\mathbf{<}\mathbf{0} \\ \mathbf{Then}\mathbf{ }{\mathbf{px}}^{\mathbf{2}}\mathbf{+}\mathbf{qx}\mathbf{+}\mathbf{r}\mathbf{>}\mathbf{0}\mathbf{ }\mathbf{for}\mathbf{ }\mathbf{all}\mathbf{ }\mathbf{x}\mathbf{\in }\mathbf{R} \\ \mathrm{In} {\mathrm{a}}^2+\mathrm{a}+1 \\ \mathrm{Coefficient} \mathrm{of} {\mathrm{a}}^2>0 \mathrm{and} \mathrm{and} \mathrm{D}=1-4=-3<0 \\ \Rightarrow {\mathrm{a}}^2+\mathrm{a}+1>0 \mathrm{for} \mathrm{all} \mathrm{a}\in \mathrm{R} \\ \Rightarrow \mathrm{Condition}\left(\mathrm{i}\right) \mathrm{will} \mathrm{not} \mathrm{be} \mathrm{satisfied} \mathrm{for} \mathrm{any} \mathrm{value} \mathrm{of} \mathrm{a}. \\ \mathbf{Hence}\mathbf{ }\mathbf{there}\mathbf{ }\mathbf{exists}\mathbf{ }\mathbf{no}\mathbf{ }\mathbf{value}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{a}\mathbf{ }\mathbf{for}\mathbf{ }\mathbf{which}\mathbf{ }\left({\mathbf{a}}^{\mathbf{2}}\mathbf{,}\mathbf{a}\right)\mathbf{ }\mathbf{and}\mathbf{ }\left(\mathbf{3}\mathbf{,}\mathbf{-}\mathbf{2}\right) \\ \mathbf{lie}\mathbf{ }\mathbf{on}\mathbf{ }\mathbf{opposite}\mathbf{ }\mathbf{side}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{line}\mathbf{ }\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{+}\mathbf{1}\mathbf{=}\mathbf{0} \end{gathered}$$

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