(2, 3, 4) বিন্দু থেকে \(\rm \dfrac{x-0}{1}=\dfrac{y-0}{0}=\dfrac{z-0}{0} \ ?\) রেখা পর্যন্ত লম্ব দূরত্ব কত?

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  1. 6 একক
  2. 5 একক
  3. 3 একক
  4. 2 একক

Answer (Detailed Solution Below)

Option 2 : 5 একক
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ধারণা:

দুটি লম্ব রেখার বিন্দু গুণফল শূন্য।

দুটি বিন্দু (x1, y1, z1) এবং (x2, y2, z2) এর মধ্যে দূরত্ব দেওয়া হয়েছে,

\(\rm \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

গণনা:

ধরা যাক, P(2, 3, 4) বিন্দু থেকে অঙ্কিত লম্বের পাদদেশ হল M

ধরি, \(\rm \dfrac{x-0}{1}=\dfrac{y-0}{0}=\dfrac{z-0}{0} =k\)

x = k, y = 0, z = 0

সুতরাং M = (k, 0, 0)

এখন PM = (2 - k, 3 - 0, 4 - 0) = (2- k, 3, 4) এবং প্রদত্ত রেখার দিক অনুপাত হল 1, 0, 0

PM প্রদত্ত রেখায় লম্ব তাই,

(2 - k) (1) + 3(0) + 4 (0) = 0

∴ k = 2

M = (2, 0, 0)

লম্ব দূরত্ব PM =

\(\rm \sqrt {(2-2)^2+(0-3)^2+(0-4)^2}\\ =\sqrt{9+16}\\ =5\)

অতএব, বিকল্প (2) সঠিক।

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