Question

The equation of the plane passing through the point $(1,2,-3)$ and perpendicular to the planes $3x+y-2z=5$ and $2x-5y-z=7$, is:

• A. $3x-10y+2z+11=0$
• B. $6x-5y-2z-2=0$
• C. $11x+y+17z+38=0$
• D. $6x-5y+2z+10=0$

Answer 1

The correct option is C

$11x+y+17z+38=0$

Form the equation of the plane based on given information

The required plane is perpendicular to the planes $3x+y-2z=5$ and $2x-5y-z=7$

Hence the normal vector to the required plane is given as

$\hat{n}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& 1& -2\\ 2& -5& -1\end{array}\right|$

$\hat{n}=\hat{i}(-11)-\hat{j}\left(1\right)+\hat{k}(-17)$

$\hat{n}=-\left(11\hat{i}+\hat{j}+17\hat{k}\right)$

Therefore, the direction ratios of the required plane are $11,1,17$

Hence the equation of the required plane can be written as

$11\left(x-1\right)+1\left(y-2\right)+17\left(z+3\right)=0$

$\Rightarrow$ $11x+y+17z+38=0$

Hence, option $\left(C\right)$ $11x+y+17z+38=0$, is the correct answer.