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Question Number 143858 by gsk2684 last updated on 19/Jun/21

$$\mathrm{if}\mathrm{A}⩾0,\mathrm{B}⩾0,\mathrm{A}+\mathrm{B}=\frac{\mathrm{\Pi }}{3}\mathrm{then}$$$$\mathrm{find}\mathrm{minimum}\mathrm{and}\mathrm{maximum}$$$$\mathrm{of}\tan \mathrm{A}.\tan \mathrm{B}$$

Answered by ajfour last updated on 19/Jun/21

$$\sqrt{3}=\frac{p+q}{1-pq}$$$$\Rightarrow 1-pq=\frac{p+q}{\sqrt{3}}$$$$pq=1-\left(\frac{p+q}{\sqrt{3}}\right)$$$$ifp=q$$$$p^2+\frac{2p}{\sqrt{3}}-1=0$$$$p=-\frac{1}{\sqrt{3}}\pm \frac{2}{\sqrt{3}}$$$$p=\frac{1}{\sqrt{3}}\Rightarrow {\left(pq\right)}_{max}=\frac{1}{3}$$$${\left(pq\right)}_{min}=0$$

Commented by ajfour last updated on 19/Jun/21

$$Arectangleofgivenperimeter$$$$hasmaximumareawhen$$$$itisasquare.$$

Commented by gsk2684 last updated on 19/Jun/21

$$\mathrm{thank}\mathrm{you}\mathrm{sir}.\mathrm{i}\mathrm{have}\mathrm{a}\mathrm{doubt}$$$$\mathrm{that}\mathrm{why}\mathrm{did}\mathrm{you}\mathrm{take}\mathrm{p}=\mathrm{q}$$

Commented by gsk2684 last updated on 21/Jun/21

$$\mathrm{thanks}\mathrm{alot}.$$

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