Question

Given the points \(A(6,-7,0),B(16,-19,-4),C(0,3,-6)\) and \(D(2,-5,10)\),the point of intersection of the lines \(AB\) and \(CD\) is 

• A. \((-1,1,2)\)
• B. \((1,-1,2)\)
• C. \((1,-1,-2)\)
• D. \((-1,1,-2)\)
• E. \((1,1,2)\)

Answer 1

The Correct Option is B

Given that

\(A(6,-7,0),B(16,-19,-4),C(0,3,-6)\) and \(D(2,-5,10)\) are the point of intersection of the lines \(AB\) and \(CD\)
So, the point f intersection can be found as ,
Any point on \(AB\) can be written as \((6+5x,−7−6x,−2x)\)
Any point on \(CD\) can be written as \((y,3−4y,−6+8y)\)
To find the intersection of  \(AB\) and \(CD\) , the coordinates of the point written in the two different ways should be equal. 
Hence, 
\(6+5x=y\)
\(−7−6x=3−4y\)
\(−2x=−6+8y\)
The three equations are consistent and on solving we get:  \(x=−1\) and \(y=1\). 
Hence, desired point of intersection is \((1,−1,2)\).
So, the correct option is (B) : \((1,−1,2)\)

Answer 2

Let lines AB and CD intersect at point P.
Let point P divide line AB in the ratio \(\lambda : 1\) and line CD in the ratio \(\mu : 1\).
Now, Coordinates of P are :
\(\left(\frac{16λ+6}{λ+1},\frac{-19λ-7}{λ+1},\frac{-4λ}{λ+1}\right)\) or \(\left(\frac{2μ}{μ+1},\frac{-5μ+3}{μ+1},\frac{10μ-6}{μ+1}\right)\)
Now, By comparing the values, we get
\(\lambda=-\frac{1}{3}\ \text{or}\ \mu=1\).
Using these value, the point of intersection can be determined as (1, -1, 2).
This also proves that lines AB and CD intersect if points A, B, C, and D are coplanar.
So, the correct option is (B) : \((1,−1,2)\)