Question

Find the equations of the plane through the point $(x_1,y_1,z_1)$ and perpendicular to the straight line $\frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}$

• A. $l(x-x_1)+m(y-y_1)+n(z-z_1)=0.$
• B. $l(x-x_1)+m(y-y_1)-n(z-z_1)=0.$
• C. $l(x-x_1)-m(y-y_1)+n(z-z_1)=0.$
• D. none of these

Answer 1

The correct option is A $l(x-x_1)+m(y-y_1)+n(z-z_1)=0.$

Since the line is perpendicular to the plane, hence the normal vector of the plane is parallel to the direction vector of the line.

Hence the equation of the plane will be

$l\left(x\right)+m\left(y\right)+n\left(z\right)=d$

It is given that the plane passes through $x_1,x_2,x_3$

Hence

$l{x}_1+m{y}_1+n{z}_1=d$

Substituting in the equation of the plane, we get

$lx+my+nz=l{x}_1+m{y}_1+n{z}_1$

Or

$l(x-x_1)+m(y-y_1)+n(z-z_1)=0$