Question

The distance of the point (7, 10, 11) from the line \( \frac{x - 4}{1} = \frac{y - 4}{0} = \frac{z - 2}{3} \) along the line \( \frac{x - 7}{2} = \frac{y - 10}{3} = \frac{z - 11}{6} \) is

• A. 18
• B. 14
• C. 12
• D. 16

Hint:

To find the distance of a point from a line along another line, assume a general point on the first line. The line joining the given point and this general point must be parallel to the second given line. Use the proportionality of direction ratios for parallel lines to find the coordinates of the point on the first line. Finally, calculate the distance between the given point and this point on the first line.

Answer 1

The Correct Option is B

\begin{center} \includegraphics{19S.png} \end{center} Let the given point be \( P(7, 10, 11) \). 
The equation of the line from which the distance is to be found is \( L_1: \frac{x - 4}{1} = \frac{y - 4}{0} = \frac{z - 2}{3} = \lambda \). 
Any point Q on this line can be written as \( Q(\lambda + 4, 0\lambda + 4, 3\lambda + 2) = (\lambda + 4, 4, 3\lambda + 2) \). 
The distance is to be found along the line \( L_2 \) passing through P and Q, whose equation is given in the solution as parallel to \( \frac{x - 7}{2} = \frac{y - 10}{3} = \frac{z - 11}{6} \). 
The direction ratios of the line PQ are \( (\lambda + 4 - 7, 4 - 10, 3\lambda + 2 - 11) = (\lambda - 3, -6, 3\lambda - 9) \). Since the line PQ is parallel to the line with direction ratios \( (2, 3, 6) \), the direction ratios of PQ must be proportional to \( (2, 3, 6) \). \[ \frac{\lambda - 3}{2} = \frac{-6}{3} = \frac{3\lambda - 9}{6} \] From \( \frac{-6}{3} = -2 \), we have: \[ \frac{\lambda - 3}{2} = -2 \Rightarrow \lambda - 3 = -4 \Rightarrow \lambda = -1 \] \[ \frac{3\lambda - 9}{6} = -2 \Rightarrow 3\lambda - 9 = -12 \Rightarrow 3\lambda = -3 \Rightarrow \lambda = -1 \] So, the value of \( \lambda \) is \( -1 \). The coordinates of the point Q on the line \( L_1 \) are \( Q(-1 + 4, 4, 3(-1) + 2) = Q(3, 4, -1) \). The distance PQ is the distance between the points \( P(7, 10, 11) \) and \( Q(3, 4, -1) \). \[ PQ = \sqrt{(7 - 3)^2 + (10 - 4)^2 + (11 - (-1))^2} \] \[ PQ = \sqrt{(4)^2 + (6)^2 + (12)^2} \] \[ PQ = \sqrt{16 + 36 + 144} \] \[ PQ = \sqrt{196} = 14 \] The distance of the point (7, 10, 11) from the line along the given line is 14.