Question

The equation of the plane perpendicular to $z$ - axis and passing through $\left(2,-3,5\right)$ is

• A. $x-2=0$
• B. $y+3=0$
• C. $z-5=0$
• D. $2x-3y+5z+4=0$

Answer 1

The correct option is C

$z-5=0$

Find the equation of the plane based on given information

The $z$ - axis is perpendicular to the required plane.

Hence, the normal to the plane is parallel to the $z$ - axis.

As the normal to the plane and $z$ - axis. are parallel the direction cosines of both are same.

The direction cosines of $z$ - axis are $\left(0,0,1\right)$

Hence, the direction cosines of normal to the plane are also $\left(0,0,1\right)$

The equation of the plane passing through the point $\left(2,-3,5\right)$ is given as

$A\left(x-2\right)+B\left(y+3\right)+C\left(z-5\right)=0$

Substitute $A=0,B=0,C=1$ we get

$0\left(x-2\right)+0\left(y+3\right)+1\left(z-5\right)=0$

$\Rightarrow$ $z-5=0$

Hence, option $\left(C\right)$, $z-5=0$ is the correct answer.