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Question:
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The planes : $2 x-y+4 z=5$ and $5 x-2.5 y+10 z=6$ are
(a) Perpendicular
(b) Parallel
(c) Intersect y-axis
(d) Passes through $\left(0,0, \frac{5}{4}\right)$
(a) Perpendicular
(b) Parallel
(c) Intersect y-axis
(d) Passes through $\left(0,0, \frac{5}{4}\right)$
Solution:
2714 Upvotes
Verified Answer
The planes $2 x-y+4 z=5$
$5 x-2.5 y+10 z=6$
Comparing the coefficients of $x, y$ and $z$
$$
\frac{2}{5}=\frac{-1}{-2.5}=\frac{4}{10}
$$
The planes $a_1 x+b_1 y+c_1 z=d_1$ and $\mathrm{a}_2 \mathrm{x}+\mathrm{b}_2 \mathrm{y}+\mathrm{c}_2 \mathrm{z}=\mathrm{d}_2$ are parallel $\frac{\mathrm{a}_1}{\mathrm{a}_2}=\frac{\mathrm{b}_1}{\mathrm{~b}_2}=\frac{\mathrm{c}_1}{\mathrm{c}_2}$ from (iii), the condition is satisified Hence planes (i) and (ii) are parallel. Option (b) is correct.
$5 x-2.5 y+10 z=6$
Comparing the coefficients of $x, y$ and $z$
$$
\frac{2}{5}=\frac{-1}{-2.5}=\frac{4}{10}
$$
The planes $a_1 x+b_1 y+c_1 z=d_1$ and $\mathrm{a}_2 \mathrm{x}+\mathrm{b}_2 \mathrm{y}+\mathrm{c}_2 \mathrm{z}=\mathrm{d}_2$ are parallel $\frac{\mathrm{a}_1}{\mathrm{a}_2}=\frac{\mathrm{b}_1}{\mathrm{~b}_2}=\frac{\mathrm{c}_1}{\mathrm{c}_2}$ from (iii), the condition is satisified Hence planes (i) and (ii) are parallel. Option (b) is correct.
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