The equation of the plane through the line of intersection of the planes $$ax+by+cz+d=0$$ and a′$$x+b$$′$$y+c$$′$$z+d$$′$$=0$$ and parallel to the line $$y=0$$ and $$z=0$$ is

Question

The equation of the plane through the line of  intersection of the planes $$ax+by+cz+d=0$$ and a′$$x+b$$′$$y+c$$′$$z+d$$′$$=0$$ and parallel to the line $$y=0$$ and $$z=0$$ is

Infinity Learn · Accepted Answer

The equation of a plane through the line of intersection of the planes ax $$+$$ by $$+$$ cz $$+$$ d $$-$$ $$0$$ and a′$$x+b$$′$$y+c$$′$$z+d$$′$$=0$$ is (ax+by+cz+d)+$$λa′$$x+b$$′$$y+c$$′$$z+d$$′$$=0or$$  $$xa+$$λa′$$+yb+$$λb′$$+zc+$$λc′$$+d+$$λd′$$=0$$ This is parallel to the $$x-axis$$, i.e., $$y=0$$,$$z=0$$. Therefore, $$1a+$$λa′$$+0b+$$λb′$$+0c+$$λc′$$=0$$ or  λ$$=$$−aa′ Putting the value of λ in $$(i), the required plane is ya′b−ab′$$+za$$′c−ac′$$+a$$′d−ad′$$=0$$ or  ab′−a′$$by+ac$$′−a′$$cz+ad$$′−a′$$d=0$$

$The equation of the plane through the line of intersection of the planes ax + by + cz + d = 0 and$
$a^{'} x + b^{'} y + c^{'} z + d^{'} = 0 and parallel to the line y = 0 and z = 0 is$

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The equation of the plane through the line of intersection of the planes ax+by+cz+d=0 and a′x+b′y+c′z+d′=0 and parallel to the line y=0 and z=0 is

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$The equation of the plane through the line of intersection of the planes ax + by + cz + d = 0 and$
$a^{'} x + b^{'} y + c^{'} z + d^{'} = 0 and parallel to the line y = 0 and z = 0 is$