Find the co-ordinates of the point where the line through the points A

1.	Find the co-ordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the xy -plane.
Answer» The Cartesian equation of line passing through two points A(3, 4, 1) and B(5, 1, 6) is \(\frac{x -3}{5-3} = \frac{y-4}{1-4} = \frac{z-1}{6-1} = \frac{y-4}{-3} = \frac{z - 1}{5}\). (By Cartesian equation of line passing through two points) Since, the line passes through points A (3,4,1) and B (5,1,6) crosses the xy - plane. Hence, z - coordinates of line is zero. Therefore, \(\frac{x-3}{2} = \frac{y-4}{-3} = \frac{0-1}{5} = -\frac{1}{5}\) ⇒ \(\frac{x-3}{2} = -\frac{1}{5}\) and \(\frac{y-4}{-3} = -\frac{1}{5}\). ⇒ x = \(-\frac{2}{5} + 3\) and y = \(\frac{3}{5} + 4 = \frac{23}{4}\). Therefore, the co-ordinates of the point is \((\frac{13}{5}, \frac{23}{5},0)\) Hence, the co-ordinates of the point where the line through the points A(3,4,1)and B(5,1,6) crosses the xy - plane is \((\frac{13}{5}, \frac{23}{5},0)\).

Find the co-ordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the xy -plane.

Answer»

The Cartesian equation of line passing through two points A(3, 4, 1) and B(5, 1, 6) is \(\frac{x -3}{5-3} = \frac{y-4}{1-4} = \frac{z-1}{6-1} = \frac{y-4}{-3} = \frac{z - 1}{5}\). (By Cartesian equation of line passing through two points)

Since, the line passes through points A (3,4,1) and B (5,1,6) crosses the xy - plane.

Hence, z - coordinates of line is zero.

Therefore, \(\frac{x-3}{2} = \frac{y-4}{-3} = \frac{0-1}{5} = -\frac{1}{5}\)

⇒ \(\frac{x-3}{2} = -\frac{1}{5}\) and \(\frac{y-4}{-3} = -\frac{1}{5}\).

⇒ x = \(-\frac{2}{5} + 3\) and y = \(\frac{3}{5} + 4 = \frac{23}{4}\).

Therefore, the co-ordinates of the point is \((\frac{13}{5}, \frac{23}{5},0)\)

Hence, the co-ordinates of the point where the line through the points A(3,4,1)and B(5,1,6) crosses the xy - plane is \((\frac{13}{5}, \frac{23}{5},0)\).

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