Question

A line joining the points \[\left( {1,2,0} \right)\] and \[\left( {4,13,5} \right)\] is perpendicular to a plane. Then find the coefficients of \[x,y\] and \[z\] in the equation of the plane.
A. \[5,15,5\]
B. \[3,11,5\]
C. \[3, - 11,5\]
D. \[ - 5, - 15,5\]

Answer 1

Hint: First, calculate the direction ratios of the line joining the points \[\left( {1,2,0} \right)\] and \[\left( {4,13,5} \right)\]. Then, use the property of the perpendicular line and plane to get the required answer.

Formula used: The direction ratios of the line joining the points \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\] is: \[\left( {{x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_1}} \right)\]

Complete step by step solution: Given:
The line joining the points \[\left( {1,2,0} \right)\] and \[\left( {4,13,5} \right)\] is perpendicular to a plane.

Let’s calculate the direction ratios of the line.
The given line joins the two points \[\left( {1,2,0} \right)\] and \[\left( {4,13,5} \right)\].
Apply the formula of the direction ratios of a line.
We get,
Direction ratios: \[\left( {1 - 4,2 - 13,0 - 5} \right)\]
\[ \Rightarrow \] Direction ratios: \[\left( { - 3, - 11, - 5} \right)\]
\[ \Rightarrow \] Direction ratios: \[\left( {3,11,5} \right)\]

It is given that the plane is perpendicular to the line joining the points \[\left( {1,2,0} \right)\] and \[\left( {4,13,5} \right)\].
So, the direction ratios of the plane and the line are equal.
We know that, the direction ratios of the line joining the points are the coefficients of \[x,y\] and \[z\] in the equation of the plane.
Thus, the required coefficients of \[x,y\] and \[z\] are \[3,11,5\].

Thus, Option (B) is correct.

Note: Students often get confused between the direction ratios and direction cosines. Remember the following formulas:
For the line joining the points \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\]:
Direction ratios: \[\left( {{x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_1}} \right)\]
Direction cosines: \[\left( {\dfrac{{{x_2} - {x_1}}}{w},\dfrac{{{y_2} - {y_1}}}{w},\dfrac{{{z_2} - {z_1}}}{w}} \right)\], where \[w = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{z_2} - {z_1}} \right)}^2}} \]