class 12

Math

3D Geometry

Three Dimensional Geometry

If the lines $1x−2 =1y−3 =−kx−4 $ and $kx−1 =2y−4 =1x−5 $are coplanar, then k can have

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Write the equation of the plane passing through the origin and parallel to the plane $5x−3y+7z+11=0$.

Find the vector and Cartesian equations of a plane which is at a distance of $7$ units from the origin and whose normal vector from the origin is $(3i^+5j^ −6k^)$.

Show that the point $(1,2,1)$ is equidistant from the planes $r⋅(i^+2j^ −2k^)=5$ and $r⋅(2i^−2j^ +k^)+3=0$.

Find the equation of the plane parallel to the plane $2x−3y+5z+7=0$ and passing through the point $(3,4,−1)$. Also, find the distance between the two planes.

Write the equations of line parallel to the line $−3x−2 =2y+3 =6z+5 $ and passing through the point $(1,−2,3)$.

Find the vector equation of the plane passing through the intersection of the planes $r⋅(2i^−7j^ +4k^)=3$ and $r⋅(3i^−5j^ +4k^)+11=0$, and passing through the point $(−2,1,3)$.

Find the coordinates of the point where the line through $(3,−4,−5)$ and $(2,−3,1)$ crosses the plane $2x+y+z=7$.

Find the equation of the plane passing through the intersection of the planes $4x−y+z=10$ and $x+y−z=4$ and parallel to the line with direction ratios $2,1,1$. Find also the perpendicular distance of $(1,1,1)$ from this plane.