Question

The distance of origin from the plane passing through points $A(3,-2,-2),B(1,2,-3)$ and $C(2,1,-1),$ is $\frac{p}{\sqrt{62}},$ the value of $p$ is

• A. $17$
• B. $19$
• C. $23$
• D. $29$

Answer 1

The correct option is B $19$

Equation of plane passing through points $A,B,C$ is:
$\left|\begin{array}{ccc}x-x_1& y-y_1& z-z_1\\ x_2-x_1& y_2-y_1& z_2-z_1\\ x_3-x_1& y_3-y_1& z_3-z_1\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}x-3& y+2& z+2\\ -2& 4& -1\\ -1& 3& 1\end{array}\right|=0\Rightarrow 7x+3y-2z-19=0$
So, perpendicular distance from origin is,
$\left|\frac{0+0+0-19}{\sqrt{7^2+3^2+2^2}}\right|=\frac{19}{\sqrt{62}}$