Question

​the region bounded by the line L bx+ay-ab=0 and the coordinate axes is a triangle in the first quadrant whose area is A. if the line passes through (1,2) then find the minimum value of A

Answer 1

Hi,
$$\begin{gathered} \mathrm{Equation}\mathrm{of}\mathrm{line}\mathrm{can}\mathrm{be}\mathrm{written}\mathrm{as}, \\ \frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1 \\ \mathrm{So}\mathrm{intersection}\mathrm{on}\mathrm{axis}\mathrm{are}\left(\mathrm{a},0\right)\mathrm{and}\left(0,\mathrm{b}\right) \\ \mathrm{So}\mathrm{area}\mathrm{of}\mathrm{triangle} \\ =\frac{1}{2}\left|\mathrm{ab}\right| \\ \mathrm{Now}\mathrm{since}\left(1,2\right)\mathrm{lies}\mathrm{on}\mathrm{line}\mathrm{so} \\ \frac{1}{\mathrm{a}}+\frac{2}{\mathrm{b}}=1 \\ 2\mathrm{a}+\mathrm{b}=\mathrm{ab} \\ \mathrm{Now}\mathrm{if}\mathrm{b}=1\mathrm{then}\mathrm{a}=-1 \\ \mathrm{So}\mathrm{minimum}\mathrm{area}\mathrm{will}\mathrm{be}=\frac{1}{2}\mathrm{unit} \end{gathered}$$

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