3d Geometry

The distance of the point (1, $-$2, 3) from the plane x $-$ y + z = 5 measured parallel to a line, whose direction ratios are 2, 3, $-$6 is :

Equation of a plane at a distance $\sqrt {{2 \over {21}}}$ from the origin, which contains the line of intersection of the planes x $-$ y $-$ z $-$ 1 = 0 and 2x + y $-$ 3z + 4 = 0, is :

The equation of the plane passing through the line of intersection of the planes $\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \right) = 1$ and $\overrightarrow r .\left( {2\widehat i + 3\widehat j - \widehat k} \right) + 4 = 0$ and parallel to the x-axis is :

Let the equation of the plane, that passes through the point (1, 4, $-$3) and contains the line of intersection of the planes 3x $-$ 2y + 4z $-$ 7 = 0 and x + 5y $-$ 2z + 9 = 0, be $\alpha$x + $\beta$y + $\gamma$z + 3 = 0, then $\alpha$ + $\beta$ + $\gamma$ is equal to :

The distance of the point ($-$1, 2, $-$2) from the line of intersection of the planes 2x + 3y + 2z = 0 and x $-$ 2y + z = 0 is :

Let ${{x - 2} \over 3} = {{y + 1} \over { - 2}} = {{z + 3} \over { - 1}}$ lie on the plane $px - qy + z = 5$, for some p, q $\in$ R. The shortest distance of the plane from the origin is :

A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\circ}$, to the $y$-axis at 45 and to the $z$-axis at an acute angle. If a plane passing through the points $(\sqrt{2},-1,1)$ and $(a, b, c)$, is normal to $\vec{v}$, then :

Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contains the line $\overrightarrow r = - \widehat k + \lambda \left( {\widehat i + \widehat j + 2\widehat k} \right),\,\lambda \in R$. Then, which of the following points lies on T?

If two distinct point Q, R lie on the line of intersection of the planes $- x + 2y - z = 0$ and $3x - 5y + 2z = 0$ and $PQ = PR = \sqrt {18}$ where the point P is (1, $-$2, 3), then the area of the triangle PQR is equal to :

Let the plane $P:\overrightarrow r \,.\,\overrightarrow a = d$ contain the line of intersection of two planes $\overrightarrow r \,.\,\left( {\widehat i + 3\widehat j - \widehat k} \right) = 6$ and $\overrightarrow r \,.\,\left( { - 6\widehat i + 5\widehat j - \widehat k} \right) = 7$. If the plane P passes through the point $\left( {2,3,{1 \over 2}} \right)$, then the value of ${{|13\overrightarrow a {|^2}} \over {{d^2}}}$ is equal to :