Question

Find the equation of the plane containing the line of intersection of the planes \[2x - y = 0\] and \[y - 3z = 0\], and perpendicular to the plane \[4x + 5y - 3z - 8 = 0\].
A. \[28x - 17y + 9z = 0\]
B. \[28x + 17y + 9z = 0\]
C. \[28x - 17y - 9z = 0\]
D. \[7x - 3y + z = 0\]

Answer 1

Hint: First, find the equation of the required plane by using the formula of the equation of the plane passing through the intersection of the two planes. After that, use the property that if two planes are perpendicular, then the sum of the products of the coefficients is zero. Then, solve the equation and calculate the value of the variable. In the end, substitute the value of the variable in the equation of the plane and get the required answer.

Formula used: The equation of the plane passing through the intersection of the two planes \[a{x_1} + b{y_1} + c{z_1} + {d_1} = 0\] and \[a{x_2} + b{y_2} + c{z_2} + {d_2} = 0\] is: \[\left( {a{x_1} + b{y_1} + c{z_1} + {d_1}} \right) + \lambda \left( {a{x_2} + b{y_2} + c{z_2} + {d_2}} \right) = 0\]
If two planes \[a{x_1} + b{y_1} + c{z_1} + {d_1} = 0\] and \[a{x_2} + b{y_2} + c{z_2} + {d_2} = 0\] are perpendicular to each other then, \[{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0\].

Complete step by step solution: Given:
The plane contains the line of intersection of the planes \[2x - y = 0\] and \[y - 3z = 0\].
The required plane is perpendicular to the plane \[4x + 5y - 3z - 8 = 0\].

Let’s calculate the equation of the required plane.
Since the plane contains the line of intersection of the planes \[2x - y = 0\] and \[y - 3z = 0\].
So, the equation of the plane is:
\[\left( {2x - y} \right) + \lambda \left( {y - 3z} \right) = 0\]
\[2x + \left( {\lambda - 1} \right)y - 3\lambda z = 0\] \[.....\left( 1 \right)\]

It is given that the required plane is perpendicular to the plane \[4x + 5y - 3z - 8 = 0\].
Use the property of the perpendicular planes \[{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0\].
We get,
\[\left( 2 \right)\left( 4 \right) + \left( {\lambda - 1} \right)\left( 5 \right) + \left( { - 3\lambda } \right)\left( { - 3} \right) = 0\]
\[ \Rightarrow 8 + 5\lambda - 5 + 9\lambda = 0\]
\[ \Rightarrow 3 + 14\lambda = 0\]
\[ \Rightarrow 14\lambda = - 3\]
\[ \Rightarrow \lambda = - \dfrac{3}{{14}}\]
Now substitute the above value in the equation \[\left( 1 \right)\]
\[2x + \left( { - \dfrac{3}{{14}} - 1} \right)y - 3\left( { - \dfrac{3}{{14}}} \right)z = 0\]
\[ \Rightarrow 2x - \dfrac{{17}}{{14}}y + \dfrac{9}{{14}}z = 0\]
Multiply both sides by \[14\].
\[ \Rightarrow 28x - 17y + 9z = 0\]
Thus, the equation of plane that contains the line of intersection of the planes \[2x - y = 0\] and \[y - 3z = 0\], and perpendicular to the plane \[4x + 5y - 3z - 8 = 0\] is \[28x - 17y + 9z = 0\].

Thus, Option (A) is correct.

Note: Students often get confused while solving the equation \[2x - \dfrac{{17}}{{14}}y + \dfrac{9}{{14}}z = 0\]. They simplify the equation as \[2x - 17y + 9z = 0 \times 14\], which is an incorrect method. So, to simplify the equation multiply both sides by \[14\]. i.e., \[14 \times 2x - 17y + 9z = 0 \times 14\].