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Question Number 55475 by Knight last updated on 25/Feb/19

$$\mathrm{Find}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{plane}$$$$\mathrm{passing}\mathrm{through}\mathrm{the}\mathrm{points}$$$$(1,0,0)\mathrm{and}(0,1,0)\mathrm{and}\mathrm{makes}$$$$\mathrm{an}\mathrm{angle}\mathrm{of}\frac{\pi }{4}\mathrm{with}\mathrm{the}\mathrm{plane}$$$$\mathrm{x}+\mathrm{y}=3$$

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Feb/19

$$leteqnofplanea(x-1)+b(y-0)+c(z-0)=0$$$$whichpassesthrough(0,1,0)$$$$a(0-1)+b(1-0)+c(0-0)=0$$$$-a+b=0[a=b]$$$$anglebetweena(x-1)+b(y-0)+c(z-0)=0$$$$andx+y+0.z=3is$$$$cos\frac{\pi }{4}=\frac{a\times 1+b\times 1+c\times 0}{\sqrt{a^2+b^2+c^2}\times \sqrt{1^2+1^2+0^2}}$$$$nos$$$$\frac{1}{\sqrt{2}}=\frac{a+b}{\sqrt{a^2+b^2+c^2}\times \sqrt{2}}$$$$1=\frac{a+a}{\sqrt{a^2+a^2+c^2}}$$$$4a^2=2a^2+c^2$$$$c=\pm a\sqrt{2}$$$$sotheeqnofplanesare$$$$a(x-1)+b(y-0)+c(z-0)=0$$$$a(x-1)+a(y-0)+a\sqrt{2}(z-0)=0$$$$x-1+y+\sqrt{2}z=0$$$$x+y+\sqrt{2}z=1$$$$and$$$$a(x-1)+b(y-0)-a\sqrt{2}(z-0)=0$$$$x-1+y-\sqrt{2}z=0$$$$x+y-\sqrt{2}z=1$$$$$$

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