Let \( (\alpha,0,\beta) \) be the coordinates of the point, which divides the line segment joining the points (1,2,-3) and (1,-1,-4) internally, then \( (\alpha^{2}+\beta^{2}) \) equals:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{130}{9} \)
Let the point divide the segment in the ratio \( m:n \). By the section formula its y-coordinate is
\( \displaystyle \frac{m(-1)+n(2)}{m+n}=0 \)
\( \displaystyle m=2n \)
So the ratio is 2 : 1. The other two coordinates are
\( \displaystyle \alpha=\frac{2(1)+1(1)}{3} \)
\( \displaystyle =1 \)
\( \displaystyle \beta=\frac{2(-4)+1(-3)}{3} \)
\( \displaystyle =-\frac{11}{3} \)
Hence
\( \displaystyle \alpha^{2}+\beta^{2}=1+\frac{121}{9} \)
\( \displaystyle =\frac{130}{9} \)