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  • Let the image of the point <img alt="\mathrm{P}(1,2,3)$ in the line $\mathrm{L}: \frac{x-6}{3}=\frac{y-1}{2}=\frac{z-2}{3}$ be $\mathrm{Q}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cm

Let the image of the point \mathrm{P}(1,2,3)$ in the line $\mathrm{L}: \frac{x-6}{3}=\frac{y-1}{2}=\frac{z-2}{3}$ be $\mathrm{Q}. Let \mathrm{R}(\alpha, \beta, \gamma) be a point that divides internally the line segment \mathrm{PQ} in the ratio 1: 3. Then the value of \mathrm{22(\alpha+\beta+\gamma)} is equal to___________

Option: 1

125


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\mathrm{Let \;M(3 a+6,2 a+1,3 a+2)\; be\; the\; mid\; point \;of \;PQ.}

\mathrm{\therefore P M \perp L} \\

\Rightarrow \overrightarrow{P M} \cdot \vec{b}=0 \\

\mathrm{\Rightarrow ((3 a+5) i+(2 a-1) j+(3 a-1) k) \cdot(3 i+2 j+3 k)=0 } \\

\mathrm{\Rightarrow a=-\frac{5}{11}} \\

\mathrm{\therefore M \text { is }\left(\frac{51}{11}, \frac{1}{11}, \frac{7}{11}\right)}

As R divides PQ in ratio 1:3, hence R is the mid-point of PM

\mathrm{R =\left(\frac{62}{22}, \frac{23}{22}, \frac{40}{22}\right)} \\

\mathrm{\therefore 22(\alpha+\beta+\gamma) =62+23+40=125}

Hence answer is 125

Posted by

manish

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