Three Dimensional Geometry

Let the image of the point $(1,0,7)$in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ be the point $(\alpha\,,\beta\,,\gamma\,)$. Then which one of the following points lies on the line passing through…

If the equation of plane passing through the mirror image of a point $\left(2,3,1\right)$with respect to line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$ and containing the line…

If the image of the point $P\left(1,-2,3\right)$in the plane, $2x+3y-4z+22=0$ measured parallel to the line, $\frac{x}{1}=\frac{y}{4}=\frac{z}{5}$ is $Q$, then $PQ$is equal to:

Consider the line $L$ given by the equation $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}.$ Let $Q$ be the mirror image of the point $(2,3,-1)$ with respect to $L.$Let a plane $P$ be such that it passes through $Q,$and the…

Let the foot of perpendicular from a point$P(1,2,-1)$ to the straight line$L:\frac{x}{1}=\frac{y}{0}=\frac{z}{-1}$ be $N$. Let a line be drawn from $P$ parallel to the plane $x+y+2z=0$ which meets $L$ at point $Q$. If…

The length of the projection of the line segment joining the points $(5,-1,4)$ and $(4,-1,3)$ on the plane, $x+y+z=7$ is

If $Q(0,-1,-3)$ is the image of the point P in the plane $3x-y+4z=2$ and$R$is the point $(3,-1,-2)$, then the area (in sq. units) of $\Delta\,PQR$ is

If the mirror image of the point $\left(1,3,5\right)$ with respect to the plane $4x-5y+2z=8$ is $\left(\alpha\,,\beta\,,\gamma\,\right)$, then$5\left(\alpha\,+\beta\,+\gamma\,\right)$ equals :

Let $Q$be the foot of perpendicular drawn from the point $P\left(1,2,3\right)$to the plane $x+2y+z=14$. If $R$is a point on the plane such that $\angle\,PRQ=60^{\circ}$, then the area of $\Delta\,PQR$ is equal to

If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3 \sqrt{5}} \mathrm{k}$, and $\int_0^{\mathrm{k}}\left[x^2\right]…

Let $P\left(-2,-1,1\right)$ and $Q\left(\frac{56}{17},\frac{43}{17},\frac{111}{17}\right)$ be the vertices of the rhombus $PRQS$. If the direction ratios of the diagonal $RS$ are $\alpha\,,-1,\beta\,$, where both…

Let a line L passing through the point $\mathrm{P}(1,1,1)$ be perpendicular to the lines $\frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1}$ and $\frac{x-17}{1}=\frac{y-71}{1}=\frac{z}{0}$. Let the line L intersect the $y z-$…