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NCERT Class XII Mathematics Chapter - - Solutions

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Question : 73 of 78
Marks: +1, -0
If the points (1, 1, p) and (–3, 0, 1) be equidistant from the plane r(3i^+4j^12k^)+13=0\vec{r} \cdot (\hat{3i} + \hat{4j} - \hat{12k}) + 13 = 0 then find the value of p.
Solution:  
The given plane is r(3i^+4j^12k^)+13=0\vec{r} \cdot (\hat{3i} + \hat{4j} - \hat{12k}) + 13 = 0
⇒ 3x + 4y – 12z + 13 = 0
Now given points are equidistant from the given plane
3×1+4×112p+139+16+144\left| \frac{3 \times 1 + 4 \times 1 - 12p + 13}{\sqrt{9+16+144}} \right| =3(3)+4(0)12(1)+139+16+144= \left| \frac{3(-3) + 4(0) - 12(1) + 13}{\sqrt{9+16+144}} \right|
⇒ |20 – 12p| = |–8|
⇒ 20 – 12p = ± 8
⇒ – 3p = 2 – 5 or – 3p = – 2 – 5
⇒ – 3p = –3 or – 3p = – 7
Hence, p=1p = 1 or p=73p = \frac{7}{3}
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