Which of the following planes are parallel but not identical?
$P_{1} : 4 x - 2 y + 6 z = 3$
$P_{2} : 4 x - 2 y - 2 z = 6$
$P_{3} : - 6 x + 3 y - 9 z = 5$
$P_{4} : 2 x - y - z = 3$

P_{2}

P_{3}

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P_{2}

P_{4}

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P_{1}

P_{3}

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P_{1}

P_{4}

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Solution

The correct option is B $P_{1}$ & $P_{3}$
a) for $P_{2}$ & $P_{3}$
Since, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \neq \frac{d_{1}}{d_{2}}$
Therefore, planes are not parallel
b) for $P_{2}$ & $P_{4}$
Since, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} = \frac{d_{1}}{d_{2}}$
Therefore, planes are identical
c) for $P_{1}$ & $P_{3}$
Since, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \neq \frac{d_{1}}{d_{2}}$
Therefore, planes are parallel and identical
d) for $P_{1}$ & $P_{4}$
Since, $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \neq \frac{d_{1}}{d_{2}}$
Therefore, planes are not parallel
Ans: C

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Which of the following planes are parallel but not identical?P1:4x−2y+6z=3P2:4x−2y−2z=6P3:−6x+3y−9z=5P4:2x−y−z=3

Which of the following planes are parallel but not identical?
$P_{1} : 4 x - 2 y + 6 z = 3$
$P_{2} : 4 x - 2 y - 2 z = 6$
$P_{3} : - 6 x + 3 y - 9 z = 5$
$P_{4} : 2 x - y - z = 3$