Question

If \( A(1,0,2) \), \( B(2,1,0) \), \( C(2,-5,3) \), and \( D(0,3,2) \) are four points and the point of intersection of the lines \( AB \) and \( CD \) is \( P(a,b,c) \), then \( a + b + c = ? \) 

• A. \( 3 \)
• B. \( -5 \)
• C. \( 5 \)
• D. \( -3 \)

Hint:

To find the intersection of two lines in 3D, parameterize each line using direction vectors, equate their coordinates, and solve for the parameters.

Answer 1

The Correct Option is A

Step 1: Find the parametric equations of line \( AB \) 
The direction ratios of line \( AB \) are given by: \[ \overrightarrow{AB} = (2 - 1, 1 - 0, 0 - 2) = (1,1,-2). \] The parametric equations of line \( AB \) are: \[ x = 1 + \lambda, \quad y = 0 + \lambda, \quad z = 2 - 2\lambda. \] 
Step 2: Find the parametric equations of line \( CD \) 
The direction ratios of line \( CD \) are given by: \[ \overrightarrow{CD} = (0 - 2, 3 + 5, 2 - 3) = (-2, 8, -1). \] The parametric equations of line \( CD \) are: \[ x = 2 - 2\mu, \quad y = -5 + 8\mu, \quad z = 3 - \mu. \] 
Step 3: Find the intersection point 
Equating \( x, y, \) and \( z \) from both parameterized equations: \[ 1 + \lambda = 2 - 2\mu, \] \[ \lambda = 3 + 2\mu. \] \[ \lambda = -5 + 8\mu. \] \[ 2 - 2\lambda = 3 - \mu. \] Solving these equations simultaneously: 1. From \( \lambda = 3 + 2\mu \) and \( \lambda = -5 + 8\mu \): \[ 3 + 2\mu = -5 + 8\mu. \] \[ 3 + 5 = 8\mu - 2\mu. \] \[ 8 = 6\mu \Rightarrow \mu = \frac{4}{3}. \] 2. Substituting \( \mu = \frac{4}{3} \) into \( \lambda = 3 + 2\mu \): \[ \lambda = 3 + 2 \times \frac{4}{3} = 3 + \frac{8}{3} = \frac{17}{3}. \] 
Step 4: Find \( a, b, c \) using parametric equations 
\[ a = 1 + \lambda = 1 + \frac{17}{3} = \frac{20}{3}. \] \[ b = 0 + \lambda = \frac{17}{3}. \] \[ c = 2 - 2\lambda = 2 - 2 \times \frac{17}{3} = 2 - \frac{34}{3} = -\frac{28}{3}. \] \[ a + b + c = \frac{20}{3} + \frac{17}{3} - \frac{28}{3} = \frac{9}{3} = 3. \] 
Final Answer: \( \boxed{3} \)