Gujarat BoardEnglish MediumSTD 12 ScienceMathsJUNE 20242 Marks
Question
Show that the line through the points $(4,7,8)$, $(2,3,4)$ is parallel to the line through the points $(-1,-2,1),(1,2,5)$.
✓
Answer
Suppose, $A (4,7,8), B (2,3,4)$, $P (-1,-2,1), Q (1,2,5)$ are given points. $\begin{array}{l} \overrightarrow{AB}=-2 \hat{i}-4 \hat{j}-4 \hat{k} \\ \overrightarrow{PQ}=2 \hat{i}+4 \hat{j}+4 \hat{k} \end{array}$ Now, $\overrightarrow{ AB }=\lambda \overrightarrow{ PQ }$ $\begin{array}{l} \therefore(-2 \hat{i}-4 \hat{j}-4 \hat{k})=\lambda(2 \hat{i}+4 \hat{j}+4 \hat{k}), \lambda \in R \\ \therefore-2=2 \lambda,-4=4 \lambda,-4=4 \lambda \\ \therefore \lambda=-1, \lambda=-1, \lambda=-1 \end{array}$ $\therefore$ Direction ratio of $\overrightarrow{ AB }$ and $\overrightarrow{ PQ }$ are equal. $\therefore$ Given both the lines are parallel.
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