Question:
The distance between the parallel planes \(4x – 2y + 4z + 9 = 0\) and \(8x – 4y + 8z + 21 = 0\) is:
समानांतर समतलों \(4x – 2y + 4z + 9 = 0\) तथा \(8x – 4y + 8z + 21 = 0\) के बीच की दूरी है:
Options:
- \(1/4\)
- \(1/2\)
- \(3/2\)
- \(7/4\)
- \(1/4\)
- \(1/2\)
- \(3/2\)
- \(7/4\)
Correct Answer: a)
Explanation:
Dividing the second equation \(8x – 4y + 8z + 21 = 0\) by 2 gives:
\[4x – 2y + 4z + \frac{21}{2} = 0\]
The distance \(d\) between two parallel planes \(ax + by + cz + d_1 = 0\) and \(ax + by + cz + d_2 = 0\) is:
\[d = \frac{|d_1 – d_2|}{\sqrt{a^2 + b^2 + c^2}}\]
Here \(d_1 = 9\), \(d_2 = 10.5\), and \(\sqrt{4^2 + (-2)^2 + 4^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6\).
\[d = \frac{|9 – 10.5|}{6} = \frac{1.5}{6} = \frac{1}{4}\]
Hence, the correct option is (a).
दूसरे समतल के समीकरण \(8x – 4y + 8z + 21 = 0\) को 2 से भाग देने पर:
\[4x – 2y + 4z + \frac{21}{2} = 0\]
दो समानांतर समतलों के बीच की दूरी का सूत्र:
\[d = \frac{|d_1 – d_2|}{\sqrt{a^2 + b^2 + c^2}}\]
यहाँ \(d_1 = 9\), \(d_2 = 10.5\), तथा \(\sqrt{4^2 + (-2)^2 + 4^2} = \sqrt{36} = 6\) है।
\[d = \frac{|9 – 10.5|}{6} = \frac{1.5}{6} = \frac{1}{4}\]
अतः सही विकल्प (a) है।