Question

If \((2,3,c)\) are the direction ratios of a ray passing through the point \(C(5,q,1)\) and also the midpoint of the line segment joining the points \(A(p,-4,2)\) and \(B(3,2,-4)\), then \(c(p+7q)=\)

• A. \(17\)
• B. \(34\)
• C. \(21\)
• D. \(28\)

Hint:

When a line passes through two points, subtract their coordinates to get its direction ratios and compare them with the given direction ratios using a proportionality constant.

Answer 1

The Correct Option is B

Step 1: Find the midpoint of \(AB\).
The given points are
\[ A(p,-4,2),\qquad B(3,2,-4) \] The midpoint \(M\) of \(AB\) is
\[ M=\left(\frac{p+3}{2},\frac{-4+2}{2},\frac{2-4}{2}\right) \] \[ M=\left(\frac{p+3}{2},-1,-1\right) \]

Step 2: Use the direction ratios of the ray.
The ray passes through
\[ C(5,q,1) \] and the midpoint
\[ M=\left(\frac{p+3}{2},-1,-1\right) \] So, direction ratios of \(CM\) are
\[ \left(\frac{p+3}{2}-5,\;-1-q,\;-1-1\right) \] \[ =\left(\frac{p-7}{2},\;-1-q,\;-2\right) \]

Step 3: Compare with given direction ratios.
Given direction ratios are
\[ (2,3,c) \] Therefore,
\[ \left(\frac{p-7}{2},-1-q,-2\right)=\lambda(2,3,c) \] So,
\[ \frac{p-7}{2}=2\lambda \] \[ -1-q=3\lambda \] and
\[ -2=c\lambda \]

Step 4: Express \(p\) and \(q\) in terms of \(c\).
From
\[ -2=c\lambda \] we get
\[ \lambda=-\frac{2}{c} \] Now,
\[ \frac{p-7}{2}=2\lambda \] \[ p-7=4\lambda \] \[ p=7+4\lambda \] \[ p=7-\frac{8}{c} \] Also,
\[ -1-q=3\lambda \] \[ q=-1-3\lambda \] \[ q=-1+\frac{6}{c} \]

Step 5: Calculate \(c(p+7q)\).
\[ p+7q=\left(7-\frac{8}{c}\right)+7\left(-1+\frac{6}{c}\right) \] \[ =7-\frac{8}{c}-7+\frac{42}{c} \] \[ =\frac{34}{c} \] Therefore,
\[ c(p+7q)=c\cdot \frac{34}{c} \] \[ =34 \]

Step 6: Final conclusion.
Hence,
\[ \boxed{34} \]