MHT CET · Maths · Straight Lines
The equation of a line passing through the point \((2,-1,1)\) and parallel to the line joining the points \(\hat{i}+2 \hat{j}+2 \hat{k}\) and \(-\hat{i}+4 \hat{j}+\hat{k}\) is
- A \(\bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(-2 \hat{i}+2 \hat{j}-\hat{k})\)
- B \(\bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(2 \hat{i}+6 \hat{j}+3 \hat{k})\)
- C \(\bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(2 \hat{i}-2 \hat{j}-\hat{k})\)
- D \(\bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(2 \hat{i}-6 \hat{j}-3 \hat{k})\)
Answer & Solution
Correct Answer
(A) \(\bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(-2 \hat{i}+2 \hat{j}-\hat{k})\)
Step-by-step Solution
Detailed explanation
Let \(\bar{b}=\hat{i}+2 \hat{j}+2 \hat{k}, \quad \bar{c}=-\hat{i}+4 \hat{j}+\hat{k}\)
\(\Rightarrow \overline{\mathrm{c}}-\overline{\mathrm{b}}=-2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}\)
The equation of the line passing through \(2 \hat{i}-\hat{j}+\hat{k}\) and parallel to \(-2 \hat{i}+2 \hat{j}-\hat{k}\) is
\(\overline{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(-2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})\)
\(\Rightarrow \overline{\mathrm{c}}-\overline{\mathrm{b}}=-2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}\)
The equation of the line passing through \(2 \hat{i}-\hat{j}+\hat{k}\) and parallel to \(-2 \hat{i}+2 \hat{j}-\hat{k}\) is
\(\overline{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(-2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})\)
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