Class 12

Math

3D Geometry

Three Dimensional Geometry

Find the equation of the plane through the points (2, 1, 1) and (1, 3, 4) and perpendicular to the plane $x2y+4z=10.$

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If the lines $−3x−1 =2ky−2 =2z−3 $ and $3kx−1 =1y−1 =−5z−6 $ are perpendicular to each other then $k=$?

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 $B(−4,7,6)$.

Find the point where the line $2x−1 =−3y−2 +4z+3 $ meets the plane $2x+4y−z=1$.

Find the vector equation of the plane passing through the point $(1,1,1)$ and parallel to the plane $r⋅(2i^−j^ +2k^)=5$.

Show that the distance of the point of intersection of the line $3x−2 =4y+1 =12z−12 $ and the plane $x−y+z=5$ from the point $(−1,−5,−10)$ is $13$ units.

Find the equation of the plane passing through the intresection of the planes $x−2y+z=1$ and $2x+y+z=8$ and parallel to the line with direction ratio proportional to $1,2,1,$ find also the perpendicular distance of $(1,1,1)$ from this plane.

For the following planes, find the direction cosines of the normal to the plane and the distance of the plane from the origin.$z=3$.

Find the equation of the plane through the line of intersection of the planes $x−3y+z+6=0$ and $x+2y+3z+5=0$, and passing through the origin.