The coordinates of the point where the line \(\displaystyle\frac{x-3}{2}=\frac{y-4}{-3}=\frac{z-1}{5}\) crosses XY plane are:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\left(\displaystyle\frac{13}{5},\frac{23}{5},0\right)\)
Let each ratio equal \(\lambda\). Then the general point of the line is
\( x=2\lambda+3,\quad y=-3\lambda+4,\quad z=5\lambda+1 \)
Every point of the XY plane has z = 0:
\( 5\lambda+1=0 \)
\( \displaystyle \lambda=-\frac{1}{5} \)
Putting this value in x and y:
\( \displaystyle x=-\frac{2}{5}+3=\frac{13}{5} \)
\( \displaystyle y=\frac{3}{5}+4=\frac{23}{5} \)
So the line crosses the XY plane at \(\left(\frac{13}{5},\frac{23}{5},0\right)\).