Question

The co-ordinates of the foot of the perpendicular drawn from the origin to a plane is \[\left( {2,4, - 3} \right)\]. Then find the equation of the plane.
A. \[2x - 4y - 3z = 29\]
B. \[2x - 4y + 3z = 29\]
C. \[2x + 4y - 3z = 29\]
D. None of these

Answer 1

Hint: First, calculate the general equation of the plane on the basis of the given point. Then, calculate the direction ratios of the plane. After that, substitute the values of the direction ratios in the equation and solve it to get the required answer.

Formula used: The equation of plane passing through the point \[\left( {{x_1},{y_1},{z_1}} \right)\] with direction ratios \[\left( {a,b,c} \right)\] is: \[a\left( {x - {x_1}} \right) + b\left( {y - {y_1}} \right) + c\left( {z - {z_1}} \right) = 0\] .

Complete step by step solution: Given: The co-ordinates of the foot of the perpendicular drawn from the origin to a plane is \[\left( {2,4, - 3} \right)\].

Let consider, \[\left( {a,b,c} \right)\] are the direction ratios of the required plane.
So, the equation of the plane passing through the point \[\left( {2,4, - 3} \right)\] is:
\[a\left( {x - 2} \right) + b\left( {y - 4} \right) + c\left( {z + 3} \right) = 0\] \[.....\left( 1 \right)\]

Since the line joining the origin and the point \[\left( {2,4, - 3} \right)\] is normal to the plane.
So, the direction ratios of normal to the plane are \[\left( {2,4, - 3} \right)\].
We get, \[a = 2,b = 4,c = - 3\]
Now substitute these values in the equation \[\left( 1 \right)\].
 \[2\left( {x - 2} \right) + 4\left( {y - 4} \right) - 3\left( {z + 3} \right) = 0\]
\[ \Rightarrow 2x - 4 + 4y - 16 - 3z - 9 = 0\]
\[ \Rightarrow 2x + 4y - 3z - 29 = 0\]
\[ \Rightarrow 2x + 4y - 3z = 29\]
Thus, the equation of the required plane is \[2x + 4y - 3z = 29\].

Thus, Option (C) 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 \[w = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{z_2} - {z_1}} \right)}^2}} \]