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Fruendlich Isotherm

Assertion :Th...

Question

Assertion :Three concurrent lines with direction cosines $l_{1}, m_{1}, n_{1}; l_{2}, m_{2}, n_{2}; l_{3}, m_{3}, n_{3}$ are coplanar if $∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$

Reason: Lines $\frac{x - x_{1}}{l_{1}} = \frac{y - y_{1}}{m_{1}} = \frac{z - z_{1}}{n_{1}}, \frac{x - x_{2}}{l_{2}} = \frac{y - y_{2}}{m_{2}} = \frac{z - z_{2}}{n_{2}}$ are coplanar if $∣ ∣ ∣ ∣ \begin{matrix} x - x_{1} & y - y_{1} & z - z_{1} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} \end{matrix} ∣ ∣ ∣ ∣ = 0$

A

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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B

Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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C

Assertion is correct but Reason is incorrect

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D

Assertion is incorrect but Reason is correct

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Solution

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
Given Three concurrent lines with direction cosines $l_{1}, m_{1}, n_{1}; l_{2}, m_{2}, n_{2}; l_{3}, m_{3}, n_{3}$ are coplanar if $∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$

Thus assertion is correct.

Lines $\frac{x - x_{1}}{l_{1}} = \frac{y - y_{1}}{m_{1}} = \frac{z - z_{1}}{n_{1}}, \frac{x - x_{2}}{l_{2}} = \frac{y - y_{2}}{m_{2}} = \frac{z - z_{2}}{n_{2}}$ are coplanar if $∣ ∣ ∣ ∣ \begin{matrix} x - x_{1} & y - y_{1} & z - z_{1} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} \end{matrix} ∣ ∣ ∣ ∣ = 0$ .

Also reason is correct but reason is not correct explanation for assertion

Thus Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Option B is right answer

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If $l_{1}, m_{1}, n_{1}, l_{2}, m_{2}, n_{2} a n d l_{3}, m_{3}, n_{3}$ are the direction cosines of three mutually perpendicular lines,then prove that the line whose direction cosines are proportional to $l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3} a n d n_{1} + n_{2} + n_{3}$ makes equal angles with them.

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