Question

If a point C with y coordinate 2, lies on the line joining the points A (-1, -4, 5) and B (4, 6, -5), then find the coordinate of C.
A) (2, 2, -1)
B) (2, 1, 2)
C) (2, 0, 0)
D) (0, 2, 4)

Answer 1

Hint: Here first we will find the equation of line AB and then satisfy the y coordinate of C and find the x and z coordinates then.
The equation of a line passing through two points \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\] is given by:-
\[\dfrac{{x - {x_2}}}{{{x_1} - {x_2}}} = \dfrac{{y - {y_2}}}{{{y_1} - {y_2}}} = \dfrac{{z - {z_2}}}{{{z_1} - {z_2}}}\]

Complete step-by-step answer:
The given endpoints of the line are:-
A (-1, 4, 5) and B (4, 6, -5)
Now we know that:-
The equation of a line passing through two points \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\] is given by:-
\[\dfrac{{x - {x_2}}}{{{x_1} - {x_2}}} = \dfrac{{y - {y_2}}}{{{y_1} - {y_2}}} = \dfrac{{z - {z_2}}}{{{z_1} - {z_2}}}\]
So for the above points
\[{x_1} = - 1,{y_1} = - 4,{z_1} = 5\]
And, \[{x_2} = 4,{y_2} = 6,{z_2} = - 5\]
Therefore, applying the formula we get:-
The equation of line AB is given by:-
\[\dfrac{{x - 4}}{{ - 1 - 4}} = \dfrac{{y - 6}}{{ - 4 - 6}} = \dfrac{{z - ( - 5)}}{{5 - ( - 5)}}\]
Solving it further we get:-
\[\dfrac{{x - 4}}{{ - 5}} = \dfrac{{y - 6}}{{ - 10}} = \dfrac{{z + 5}}{{10}}\]
Putting this equation equal to constant k we get:-
\[\dfrac{{x - 4}}{{ - 5}} = \dfrac{{y - 6}}{{ - 10}} = \dfrac{{z + 5}}{{10}} = k\]
Now equating first and fourth term we get:-
\[\dfrac{{x - 4}}{{ - 5}} = k\]
Solving for x we get:-
\[x - 4 = - 5k\]
\[ \Rightarrow x = - 5k + 4\]…………………………… (1)
Now equating third and fourth term we get:-
\[\dfrac{{z + 5}}{{10}} = k\]
Solving for z we get:-
\[z + 5 = 10k\]
\[ \Rightarrow z = 10k - 5\]……………………………….. (2)
Now equating second and fourth term we get:-
\[\dfrac{{y - 6}}{{ - 10}} = k\]
Solving for y we get:-
\[y - 6 = - 10k\]
\[ \Rightarrow y = - 10k + 6\]………………………….. (3)
Now we since point C lies on the line AB
Therefore, the coordinates of C should satisfy the equation of AB
Now since we know that y coordinate of C is 2
Hence substituting this value in equation 3 we get:-
\[2 = - 10k + 6\]
Solving for k we get:-
\[ - 10k = 2 - 6\]
\[ \Rightarrow k = \dfrac{{ - 4}}{{ - 10}}\]
\[ \Rightarrow k = \dfrac{2}{5}\]
Putting this value in equation1 we get:-
\[x = - 5\left( {\dfrac{2}{5}} \right) + 4\]
Solving for x we get:-
\[
   \Rightarrow x = - 2 + 4 \\
   \Rightarrow x = 2 \\
 \]
Now putting the value of k in equation 2 we get:-
\[z = 10\left( {\dfrac{2}{5}} \right) - 5\]
Solving for z we get:-
\[
  z = 4 - 5 \\
   \Rightarrow z = - 1 \\
 \]
Therefore the coordinates of C are (2, 2, -1)

Hence option A is correct.

Note: Students should take a note that when a point lies on a curve or a line then its coordinates should satisfy the equation of the curve or the line.