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Question Number 32838 by Rio Mike last updated on 03/Apr/18

the points M(1,1),N(2,4) andR (3 ,q) lie on theresame straight line. find the coordinates of R

$${the}\:{points}\:{M}\left(\mathrm{1},\mathrm{1}\right),{N}\left(\mathrm{2},\mathrm{4}\right)\:{andR}\:\left(\mathrm{3}\:,{q}\right) \\ $$$${lie}\:{on}\:{theresame}\:{straight}\:{line}. \\ $$$${find}\:{the}\:{coordinates}\:{of}\:{R} \\ $$

Commented by Rasheed.Sindhi last updated on 03/Apr/18

M(2,4)=N(2,4) is one point.So there are only two points!

$${M}\left(\mathrm{2},\mathrm{4}\right)={N}\left(\mathrm{2},\mathrm{4}\right)\:\mathrm{is}\:\mathrm{one}\:\mathrm{point}.\mathrm{So}\:\mathrm{there}\:\mathrm{are} \\ $$$$\mathrm{only}\:\mathrm{two}\:\mathrm{points}!\: \\ $$

Commented by Rio Mike last updated on 03/Apr/18

please redo question was wrong

$${please}\:{redo}\:{question}\:{was}\:{wrong} \\ $$

Answered by MJS last updated on 03/Apr/18

y=kx+d for M: 1=1k+d ⇒ d=1−k (I) for N: 4=2k+d (II) I in II: 4=2k+1−k ⇒ k=3 ⇒ d=−2 y=3x−2 for R: y=3×3−2=7

$${y}={kx}+{d} \\ $$$$\mathrm{for}\:{M}:\:\mathrm{1}=\mathrm{1}{k}+{d}\:\Rightarrow\:{d}=\mathrm{1}−{k}\:\left(\mathrm{I}\right) \\ $$$$\mathrm{for}\:{N}:\:\mathrm{4}=\mathrm{2}{k}+{d}\:\left(\mathrm{II}\right) \\ $$$$\mathrm{I}\:\mathrm{in}\:\mathrm{II}:\:\mathrm{4}=\mathrm{2}{k}+\mathrm{1}−{k}\:\Rightarrow\:{k}=\mathrm{3}\:\Rightarrow\:{d}=−\mathrm{2} \\ $$$${y}=\mathrm{3}{x}−\mathrm{2} \\ $$$$\mathrm{for}\:\mathrm{R}:\:{y}=\mathrm{3}×\mathrm{3}−\mathrm{2}=\mathrm{7} \\ $$

Answered by Rasheed.Sindhi last updated on 04/Apr/18

AnOther method Slope of MN^(−) =Slope of NR^(−) [∵ MN^(−) or NR^(−) is same line] ((4−1)/(2−1))=((q−4)/(3−2)) q−4=3 q=7

$$\mathrm{AnOther}\:\mathrm{method} \\ $$$$\mathrm{Slope}\:\mathrm{of}\:\overline {\mathrm{MN}}=\mathrm{Slope}\:\mathrm{of}\:\overline {\mathrm{NR}}\:\left[\because\:\overline {\mathrm{MN}}\:\mathrm{or}\:\overline {\mathrm{NR}}\:\mathrm{is}\:\mathrm{same}\:\mathrm{line}\right] \\ $$$$\:\:\:\:\:\:\frac{\mathrm{4}−\mathrm{1}}{\mathrm{2}−\mathrm{1}}=\frac{\mathrm{q}−\mathrm{4}}{\mathrm{3}−\mathrm{2}} \\ $$$$\:\:\:\:\:\mathrm{q}−\mathrm{4}=\mathrm{3} \\ $$$$\:\:\:\:\:\:\mathrm{q}=\mathrm{7} \\ $$

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