If the line y - (3)^1/2x + 3 = 0 cuts the parabola y² = x + 2 at A and B, and if P = (3^1/2, 0), then PA.PB is equal to?

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{y}-0=\sqrt{3}\left(\mathrm{x}-\sqrt{3}\right) \\ \mathrm{y}=\sqrt{3}\mathrm{x}-3 \\ {\mathrm{y}}^2=\mathrm{x}+2 \\ {\left(\sqrt{3}\mathrm{x}-3\right)}^2=\mathrm{x}+2 \\ 3{\mathrm{x}}^2+9-6\sqrt{3}\mathrm{x}=\mathrm{x}+2 \\ 3{\mathrm{x}}^2-\mathrm{x}\left(6\sqrt{3}+1\right)+7=0 \\ \mathrm{x}1+\mathrm{x}2=\frac{6\sqrt{3}+1}{3}----\left(1\right) \\ \mathrm{x}1.\mathrm{x}2=\frac{7}{3}---\left(2\right) \\ \mathrm{Now}\mathrm{to}\mathrm{solve}\mathrm{the}\mathrm{problem} \\ \mathrm{we}\mathrm{take}\mathrm{PA}\mathrm{and}\mathrm{PB} \\ \mathrm{any}\mathrm{point}\mathrm{on}\mathrm{parabola}\mathrm{can}\mathrm{be}\mathrm{written}\mathrm{as},\left(\mathrm{x},\sqrt{\mathrm{x}+2}\right) \\ \mathrm{let}\mathrm{A}\left(\mathrm{x}1,\sqrt{\mathrm{x}1+2}\right) \\ \mathrm{B}\left(\mathrm{x}2,\sqrt{\mathrm{x}2+2}\right) \\ \mathrm{PA}=\sqrt{{\left(\sqrt{3}-\mathrm{x}1\right)}^2+{\left(\sqrt{\mathrm{x}1+2}\right)}^2}=\sqrt{3+\mathrm{x}{1}^2-2\sqrt{3}\mathrm{x}1+\mathrm{x}1+2} \\ \mathrm{PB}=\sqrt{{\left(\sqrt{3}-\mathrm{x}2\right)}^2+{\left(\sqrt{\mathrm{x}2+2}\right)}^2}=\sqrt{3+\mathrm{x}{2}^2-2\sqrt{3}\mathrm{x}2+\mathrm{x}2+2} \\ \mathrm{PA}\times \mathrm{PB}=\sqrt{\left(5+\mathrm{x}{1}^2-2\sqrt{3}\mathrm{x}1+\mathrm{x}1\right)\left(5+\mathrm{x}{2}^2-2\sqrt{3}\mathrm{x}2+\mathrm{x}2\right)} \\ =\sqrt{25+5\mathrm{x}{1}^2+5\mathrm{x}{2}^2-5\mathrm{x}1\left(2\sqrt{3}-1\right)-5\mathrm{x}2\left(2\sqrt{3}-1\right)+{\left(\mathrm{x}1\mathrm{x}2\right)}^2-\mathrm{x}1.\mathrm{x}{2}^2\left(2\sqrt{3}-1\right)-\mathrm{x}{2}^2.\mathrm{x}1\left(2\sqrt{3}-1\right)+\mathrm{x}1\mathrm{x}2{\left(2\sqrt{3}-1\right)}^2} \\ \mathrm{all}\mathrm{values}\mathrm{in}\mathrm{above}\mathrm{equation}\mathrm{can}\mathrm{be}\mathrm{calculated}\mathrm{using}1\mathrm{and}2\mathrm{above}. \\ \mathrm{calculations}\mathrm{are}\mathrm{complex}. \\ \mathrm{the}\mathrm{final}\mathrm{value}\mathrm{should}\mathrm{be}\mathrm{equal}\mathrm{to}\frac{4}{3}\left(2+\sqrt{3}\right) \end{gathered}$$

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