Class 12

Math

3D Geometry

Three Dimensional Geometry

Show that the equation $ax+by+d=0$ represents a plane parallel to the z-axis. Hence, find the equation of a plane which is parallel to the z-axis and passes through the points $A(2,−3,1)$ and $B(−4,7,6)$.

The given equation is $ax+by+0.z+d=0$ which is of the form $ax+by+cz+d=0$.

Therefore, it represents a plane.

Direction ratio of normal to the plane are $a,b,0$

Direction ratio of the z-axis are $0,0,1$

Now, $a×0+b×0+0×1=0$

This shows that the given plane is parallel to the z-axis.

Let the required plane be $ax+by+d=0$ ..........(i)

Since it passes through the points $A(2,−3,1)$ and $B(−4,7,6)$, we have

$2a−3b+d=0$ ........(ii)

$−4a+7b+d=0$ .........(iii)

On solving (ii) and (iii) by cross multiplication, we get

$(−3−7)a =(−4−2)b =(14−12)c $

$⇒−10a =−6b =2c ⇒5a =3b =−1c =k$ (say)

$∴a=5k,b=3k$ and $c=−k$

Putting these values in (i), we get

$5kx+3ky−k=0$ $⇒5x+3y−1=0$,

which is the required equation of the plane.