Let the image of the point (1,0,7) in the line x1=y-12=z-23 be the point (,,).

Question

Let the image of the point $(1,0,7)$ in the line $\dfrac{x}{1}=\dfrac{y-1}{2}=\dfrac{z-2}{3}$ be the point $(\alpha,\beta,\gamma)$. Then which one of the following points lies on the line passing through $(\alpha,\beta,\gamma)$ and making angles $\dfrac{2\pi}{3}$ and $\dfrac{3\pi}{4}$ with the $y$-axis and $z$-axis respectively, and an acute angle with the $x$-axis?

Accepted Answer

$$\mathrm{L}_1=\frac{\mathrm{x}}{1}=\frac{\mathrm{y}-1}{2}=\frac{\mathrm{z}-2}{3}=\lambda$$

$$\begin{aligned}
& \mathrm{M}(\lambda, 1+2 \lambda, 2+3 \lambda) \
& \overrightarrow{\mathrm{PM}}=(\lambda-1) \hat{\mathrm{i}}+(1+2 \lambda) \hat{\mathrm{j}}+(3 \lambda-5) \hat{\mathrm{k}}
\end{aligned}$$

$$\overrightarrow{\mathrm{PM}}$$ is perpendicular to line $$\mathrm{L}_1$$

$$\begin{aligned}
& \overrightarrow{\mathrm{PM}} \overrightarrow{\mathrm{b}}=0 \quad(\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}) \
& \Rightarrow \lambda-1+4 \lambda+2+9 \lambda-15=0 \
& 14 \lambda=14 \Rightarrow \lambda=1 \
& \therefore \mathrm{M}=(1,3,5) \
& \overrightarrow{\mathrm{Q}}=2 \overrightarrow{\mathrm{M}}-\overrightarrow{\mathrm{P}}[\mathrm{M} \text { is midpoint of } \overrightarrow{\mathrm{P}} & \overrightarrow{\mathrm{Q}}]
\end{aligned}$$

$$\begin{aligned}
& \overrightarrow{\mathrm{Q}}=2 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+10 \hat{\mathrm{k}}-\hat{\mathrm{i}}-7 \hat{\mathrm{k}} \
& \overrightarrow{\mathrm{Q}}=\hat{\mathrm{i}}+6 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \
& \therefore(\alpha, \beta, \gamma)=(1,6,3)
\end{aligned}$$

Required line having direction cosine $$(l, \mathrm{~m}, \mathrm{n})$$

$$\begin{aligned}
& l^2+m^2+n^2=1 \
& \Rightarrow l^2+\left(-\frac{1}{2}\right)^2+\left(-\frac{1}{\sqrt{2}}\right)^2=1
\end{aligned}$$

$$l^2=\frac{1}{4}$$

$$\therefore l=\frac{1}{2}$$ [Line make acute angle with $$\mathrm{x}$$-axis]

Equation of line passing through $(1,6,3)$ will be

$$\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+6 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})+\mu\left(\frac{1}{2} \hat{\mathrm{i}}-\frac{1}{2} \hat{\mathrm{j}}-\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\right)$$

Option (3) satisfying for $$\mu=4$$

Solutions · 1