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Question Number 200937 by Mingma last updated on 26/Nov/23

Answered by MM42 last updated on 26/Nov/23

$$\frac{25\sqrt{2}}{4}\pi$$

Commented by Mingma last updated on 26/Nov/23

Please show workings

Commented by Mingma last updated on 26/Nov/23

Nice solution!

Answered by witcher3 last updated on 26/Nov/23

$$\mathrm{z}=\mathrm{x}+\mathrm{iy};$$$$\iff \sqrt{{\mathrm{x}}^2+{(\mathrm{y}-3)}^2}+\sqrt{{(\mathrm{x}-4)}^2+{\mathrm{y}}^2}⩽5\sqrt{2}.....\mathrm{R}$$$$\mathrm{E}=\{(\mathrm{x},\mathrm{y})\mid \mathrm{R}\mathrm{is}\mathrm{True}\}$$$$\mathrm{MF}+{\mathrm{MF}}^{\prime }=2\mathrm{a}\mathrm{defind}\mathrm{elpise}$$$$\mathrm{MF}+{\mathrm{MF}}^{\prime }⩽2\mathrm{a}\mathrm{all}\mathrm{point}\mathrm{inside}\mathrm{elipse}$$$$2\mathrm{a}=5\sqrt{2}\Rightarrow \mathrm{a}=\frac{5}{\sqrt{2}}$$$$\mathrm{F}(0,3),{\mathrm{F}}^{\prime }(4,0)$$$${\mathrm{FF}}^{\prime }=5$$$$\mathrm{c}=\frac{5}{2}$$$$\mathrm{b}=\sqrt{{\mathrm{a}}^2-{\mathrm{c}}^2}=\sqrt{\frac{25}{2}-\frac{25}{4}}=\frac{5}{2}$$$$\mathrm{S}=\int {\int }_{\mathrm{E}}\mathrm{dxdy}=\int {\int }_{\mathrm{E}}{1}_{\mathrm{E}}\mathrm{dxdy}={\mathrm{S}}_{\left(\mathrm{elipse}\right)}=\pi .\frac{5}{\sqrt{2}}.\frac{5}{2}$$$${\mathrm{S}}_{\mathrm{elipse}}=\pi \mathrm{ab}$$$$=\pi \frac{25\sqrt{2}}{4}$$$$$$

Commented by Mingma last updated on 26/Nov/23

Nice solution!

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