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Question Number 194903 by cortano12 last updated on 19/Jul/23

Commented by cortano12 last updated on 19/Jul/23

$$x=\sqrt{39}\left(\times \right)$$$$x=\sqrt{30}\left(✓\right)$$

Answered by MM42 last updated on 19/Jul/23

$$tan(\alpha +\theta )=\frac{10}{x}$$$$\frac{tan\alpha +tan\theta }{1-tan\alpha \times tan\theta }=\frac{10}{x}$$$$\frac{\frac{3}{x}+tan\theta }{1-\frac{3}{x}tan\theta }=\frac{10}{x}\Rightarrow 3x+x^{2}tan\theta =10x-30tan\theta$$$$\Rightarrow tan\theta =\frac{7x}{x^2+30}\Rightarrow \theta ={tan}^{-1}\left(\frac{7x}{x^2+30}\right)$$$$\Rightarrow {\theta }^{{}^{\prime }}=\frac{\frac{7(x^2+30)-2x\left(7x\right)}{{(x^2+30)}^2}}{1+{(x^2+30)}^2}=0$$$$\Rightarrow x=\sqrt{30}✓$$$$$$

Commented by MM42 last updated on 19/Jul/23

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