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QJEEM24Mabb7c: JEE Main 2024 Mathematics - Class 12 JEEM24Mabb7c Solution: A line passes through and . The point , where are non-negative…

A line passes through A(4,−6,−2)A(4,-6,-2) and B(16,−2,4)B(16,-2,4) . The point P(a,b,c)P(a, b, c) , where a,b,ca, b, c are non-negative integers, on the line ABA B lies at a distance of 21 units, from the point AA . The distance between the points P(a,b,c)P(a, b, c) and Q(4,−12,3)Q(4,-12,3) is equal to __________.

Step-by-Step Verified Solution

amp;x−412=x+64=z+26amp;x−467=y+627=z+237=21amp;(21×67+4,27×21−6,37×21−2)amp;=(22,0,7)=(a,b,c)amp;∴324+144+16=22\begin{aligned}& \frac{x-4}{12}=\frac{x+6}{4}=\frac{z+2}{6} \\& \frac{x-4}{\frac{6}{7}}=\frac{y+6}{\frac{2}{7}}=\frac{z+2}{\frac{3}{7}}=21 \\& \left(21 \times \frac{6}{7}+4, \frac{2}{7} \times 21-6, \frac{3}{7} \times 21-2\right) \\& =(22,0,7)=(a, b, c) \\& \therefore \sqrt{324+144+16}=22\end{aligned}

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