If (2 +1, ^2-3 ,( -1)/(2)) is the image of ( ,2 ,1) in the line (x-2)/(3)=(y-1)/(2)=(z)/(1…

Question

If $(2\alpha +1,{\alpha }^{2}-3\alpha ,\frac{\alpha -1}{2})$ is the image of $(\alpha ,2\alpha ,1)$ in the line $\frac{x-2}{3}=\frac{y-1}{2}=\frac{z}{1}$, then the possible value(s) of $\alpha $ is (are)

Accepted Answer

$	herefore P$ is mid point of AB and lies on the given line

$\Rightarrow P(\frac{3\alpha +1}{2},\frac{{\alpha }^{2}-\alpha }{2},\frac{\alpha +1}{4})$

Now put coordinates of P in the line

$\begin{gathered} \Rightarrow \frac{\frac{3\alpha +1}{2}-2}{3}=\frac{\frac{{\alpha }^{2}-\alpha }{2}-1}{2}=\frac{\alpha +1}{4} \ \Rightarrow \frac{\alpha -1}{2}=\frac{\alpha +1}{4}=\frac{{\alpha }^{2}-\alpha -2}{4} \ \Rightarrow \alpha =3 \end{gathered}$

Now for $\alpha =3$, clearly AB is perpendicular to $3\hat{i}+2\hat{j}+k$

for $\alpha =3$ point B is image of point A

Solutions · 1