Question

The equation of the plane perpendicular to the line \[\dfrac{x+2}{1}=\dfrac{y-1}{2}=\dfrac{z+1}{-1}\] and passing through the point (3,0,5) is \[Ax\text{ }+\text{ }By\text{ }+\text{ }Cz\text{ }+\text{ }2\text{ }=\text{ }0\], then \[\left( A\text{ }+\text{ }B\text{ }+\text{ }C \right)\]is, choose the correct option.
A. 2
B. -2
C. 4
D. 0

Answer 1

Hint: When the line equation is given then the direction ratio will be parallel to the plane that is perpendicular to the given line. So, the plane equation will be of the form \[Ax\text{ }+\text{ }By\text{ }+\text{ }Cz\text{ }+\text{ }D\text{ }=\text{ }0\], where A, B and C are the direction ratio of the given line \[\dfrac{x+2}{1}=\dfrac{y-1}{2}=\dfrac{z+1}{-1}\]. Then we can compare the equation so formed with the given \[Ax\text{ }+\text{ }By\text{ }+\text{ }Cz\text{ }+\text{ }2\text{ }=\text{ }0\] to find A, B and C and then find the sum \[\left( A\text{ }+\text{ }B\text{ }+\text{ }C \right)\].

Complete step-by-step answer:
In the question, it is given that the equation of the plane perpendicular to the line \[\dfrac{x+2}{1}=\dfrac{y-1}{2}=\dfrac{z+1}{-1}\] and passing through the point (3,0,5) is \[Ax\text{ }+\text{ }By\text{ }+\text{ }Cz\text{ }+\text{ }2\text{ }=\text{ }0\].
So, we know that the plane that is perpendicular to the given line will have the normal to the plane as parallel to the given line equation This means that the normal to the plane will have the same direction ratio as the direction ratio of the line. So, the direction ratio of the line \[\dfrac{x+2}{1}=\dfrac{y-1}{2}=\dfrac{z+1}{-1}\]are \[A=1\], \[B=2\]and \[C=-1\]. And this will be the same as the direction ratio of the plane of the form \[Ax\text{ }+\text{ }By\text{ }+\text{ }Cz\text{ }+\text{ }D\text{ }=\text{ }0\]. Now, when we put the values of A, B and C in this plane equation we will get the plane \[x\text{ }+2y\text{ }-z\text{ }+\text{ }D\text{ }=\text{ }0\]. Now, we have to compare this plane equation with the given plane equation \[Ax\text{ }+\text{ }By\text{ }+\text{ }Cz\text{ }+\text{ }2\text{ }=\text{ }0\]. So, here we get \[A=1\], \[B=2\]and \[C=-1\].
So, now we have to find the value of \[\left( A\text{ }+\text{ }B\text{ }+\text{ }C \right)\]is found as follows:
\[\begin{align}
  & \Rightarrow \left( A\text{ }+\text{ }B\text{ }+\text{ }C \right)=\left( \text{1 }+\text{ }2\text{ -1} \right) \\
 & \Rightarrow \left( A\text{ }+\text{ }B\text{ }+\text{ }C \right)=\left( 2 \right) \\
\end{align}\]
So, the required value is 2. Hence, the correct answer is option A.

Note: It can be noted that the line equation should be in the simplest form, then only we take the denominator as the direction ratios as the direction ratios of the plane of the form \[Ax\text{ }+\text{ }By\text{ }+\text{ }Cz\text{ }+\text{ }D\text{ }=\text{ }0\]. Also, it is not necessary to find the value of D , as we are not finding the sum of that.