Question 11 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: If the cartesian equation of a line is $\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2},$ then its vector form is $\vec{\text{r}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}+\lambda(3\hat{\text{i}}+7\hat{\text{j}}+2\hat{\text{k}}).$
Reason: The cartesian equation of the line which passes through the point (-2, 4, -5) and parallel to the line given by $\frac{\text{x}+3}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}+8}{6}$ is $\frac{\text{x}+3}{-2}=\frac{\text{y}-4}{4}=\frac{\text{z}+8}{-5}.$
Assertion: If the cartesian equation of a line is $\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2},$ then its vector form is $\vec{\text{r}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}+\lambda(3\hat{\text{i}}+7\hat{\text{j}}+2\hat{\text{k}}).$
Reason: The cartesian equation of the line which passes through the point (-2, 4, -5) and parallel to the line given by $\frac{\text{x}+3}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}+8}{6}$ is $\frac{\text{x}+3}{-2}=\frac{\text{y}-4}{4}=\frac{\text{z}+8}{-5}.$
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Answer
In assertion the given cartesian equation is
$\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2},$
$\Rightarrow\vec{\text{a}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}$ and $\vec{\text{b}}=3\hat{\text{i}}+7\hat{\text{j}}+2\hat{\text{k}}$
The vector equation of the line is given by $\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}},\lambda\in\text{R}.$
$\Rightarrow\vec{\text{r}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}+\lambda(3\hat{\text{i}}+7\hat{\text{j}}+2\text{k})$
Thus Assertion is correct. In reason it is given that the line passes through the point (-2, 4, -5) and is parallel to
Clearly, the direction ratios of line are (3, 5, 6). Now the equation of the line (in cartesian form) is
$\frac{\text{x}-(-2)}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}-(-5)}{6}$
$\Rightarrow\frac{\text{x}+2}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}+5}{6}$
View full question & answer→- Assertion is correct statement but Reason is wrong statement.
In assertion the given cartesian equation is
$\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2},$
$\Rightarrow\vec{\text{a}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}$ and $\vec{\text{b}}=3\hat{\text{i}}+7\hat{\text{j}}+2\hat{\text{k}}$
The vector equation of the line is given by $\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}},\lambda\in\text{R}.$
$\Rightarrow\vec{\text{r}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}+\lambda(3\hat{\text{i}}+7\hat{\text{j}}+2\text{k})$
Thus Assertion is correct. In reason it is given that the line passes through the point (-2, 4, -5) and is parallel to
Clearly, the direction ratios of line are (3, 5, 6). Now the equation of the line (in cartesian form) is
$\frac{\text{x}-(-2)}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}-(-5)}{6}$
$\Rightarrow\frac{\text{x}+2}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}+5}{6}$