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Question Number 95342 by i jagooll last updated on 24/May/20

$$\mathrm{determine}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{k}\mathrm{such}\mathrm{that}$$$$\mathrm{the}\mathrm{point}\mathrm{A}(4,-2,6)\mathrm{B}(0,1,0)\mathrm{C}(1,0,-5)$$$$\mathrm{and}\mathrm{D}(1,\mathrm{k},-2)\mathrm{lie}\mathrm{on}\mathrm{the}\mathrm{same}$$$$\mathrm{plane}$$

Answered by john santu last updated on 24/May/20

$$\mathrm{let}\mathbf{u}=\mathbf{A}\mathrm{B}=(-4,3,-6)$$$$\mathbf{v}=(-3,2,-11).\mathrm{let}\mathbf{n}\mathrm{vector}$$$$\mathrm{normal}\mathbf{n}=\mathbf{u}\times \mathbf{v}=\left|\begin{array}{c}-43-6\\ -32-11\end{array}\right|$$$$\mathbf{n}=(-21,-26,1).\mathrm{set}\mathrm{P}(\mathrm{x},\mathrm{y},\mathrm{z})$$$$\mathrm{arbitary}\mathrm{point}\mathrm{in}\mathrm{plane}.$$$$\mathrm{put}\mathrm{BP}=(\mathrm{x},\mathrm{y}-1,\mathrm{z})$$$$\mathrm{then}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{plane}\mathrm{is}$$$$\mathbf{n}.\mathbf{B}\mathrm{P}=0$$$$-21\mathrm{x}-26(\mathrm{y}-1)+\mathrm{z}=0$$$$\mathrm{so}\mathrm{take}\mathrm{x}=1,\mathrm{y}=\mathrm{k}\mathrm{and}\mathrm{z}=-2$$$$-21.1-26(\mathrm{k}-1)-2=0$$$$-21-26\mathrm{k}+26-2=0$$$$26\mathrm{k}=3\Rightarrow \mathrm{k}=\frac{3}{26}.$$

Commented by i jagooll last updated on 24/May/20

$$\mathrm{thank}\mathrm{you}$$

Answered by mr W last updated on 24/May/20

$$\left|\begin{array}{ccc}4& -3& 6\\ 1& -1& -5\\ 1& k-1& -2\end{array}\right|=0$$$$\left|\begin{array}{ccc}4& -3& 6\\ 0& 1& 26\\ 0& k& 3\end{array}\right|=0$$$$4(3-26k)=0$$$$\Rightarrow k=\frac{3}{26}$$

Commented by i jagooll last updated on 24/May/20

$$\mathrm{thank}\mathrm{you}$$

Commented by john santu last updated on 24/May/20

$$\mathrm{how}\mathrm{you}\mathrm{get}(1,-1,-5)\mathrm{and}$$$$(1,\mathrm{k}-1,2)?$$

Commented by PRITHWISH SEN 2 last updated on 24/May/20

$$\mathrm{by}\mathrm{substracting}\mathrm{points}$$

Commented by john santu last updated on 24/May/20

$$\mathrm{ooo}\mathrm{thank}\mathrm{you}$$

Commented by mr W last updated on 24/May/20

$$vectorsAB,CBandDBareinthe$$$$sameplane.$$

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