Question

Let P be the point (10, –2, –1) and Q be the foot of the perpendicular drawn from the point R(1, 7, 6) on the line passing through the points (2, –5, 11) and (–6, 7, –5). Then the length of the line segment PQ is equal to ________.

Answer 1

Correct answer is: 13

Line : \(\frac{x+6}{-8}=\frac{y-7}{12}=\frac{z+5}{-16}\)

\(\frac{\mathrm{x}+6}{2}=\frac{\mathrm{y}-7}{-3}=\frac{\mathrm{z}+5}{4}=\lambda\)

\(\mathrm{Q}(2 \lambda-6,7-3 \lambda, 4 \lambda-5)\)

\(\overline{\mathrm{QR}}(2 \lambda-7,-3 \lambda, 4 \lambda-11)\)

\(\overline{\mathrm{QR}} \) . dr's of line = 0

\(4 \lambda-14+9 \lambda+16 \lambda-44=0\)

\(29 \lambda=58 \Rightarrow \lambda=2\)

\(\mathrm{Q}(-2,1,3)\)

\(\mathrm{PQ}=\sqrt{144+9+16}=\sqrt{169}=13\)