Three Dimensional Geometry

Paragraph : Consider the spheres $x^2+y^2$ $+z^2-4 y+3=0$ and $x^2+y^2+z^2+2 x+4 z-4=0$. Question : What is the distance between the centres of the two spheres?

What is the diameter of the sphere $x^2+y^2+z^2-4 x+$ $6 y-8 z-7=0$

Coordinate planes and the planes $\pi_1, \pi_2, \pi_3$ which are respectively parallel to $\mathrm{YZ}, \mathrm{ZX}, \mathrm{XY}$ planes at distances $\mathrm{a}, \mathrm{b}, \mathrm{c}$, form a rectangular…

The values of $x$ for which the angle between the vectors $x^2 \hat{i}+2 x \hat{j}+\hat{k}$ and $\hat{i}-2 \hat{j}+x \hat{k}$ is obtuse, lie in the interval

In $\triangle \mathrm{ABC}$, if the midpoints of the sides $\mathrm{AB}, \mathrm{BC}$ and $\mathrm{CA}$ are respectively $(l, 0,0),(0, \mathrm{~m}, 0)$ and $(0,0, \mathrm{n})$, then…

In a quadrilateral $A B C D$, the point $P$ divides $D C$ in the ratio $1: 2$ and $Q$ is the mid point of $A C$. If $\overrightarrow{\mathbf{A B}}+2 \overrightarrow{\mathbf{A D}}+\overrightarrow{\mathbf{B C}}-2…

$\triangle A B C$ is formed by $A(1,8,4), B(0,-11,4)$ and $C(2,-3,1)$. If $D$ is the foot of the perpendicular from $A$ to $B C$. Then the coordinates of $D$ are

In $\triangle A B C$ the mid points of the sides $A B, B C$ and $C A$ are respectively $(l, 0,0),(0, m, 0)$ and $(0,0, n)$. Then, $\frac{A B^2+B C^2+C A^2}{l^2+m^2+n^2}$ is equal to

The equation of a plane passing through $(-1,2,3)$ and whose normal makes equal angles with the coordinate axes is

If $l, m, n$ are the direction cosines of a line that is perpendicular to the lines having the direction ratios $1,2,-1$ and $1,-2,1$ then $(l+m+n)^2=$

If the foot of the perpendicular drawn from $(0,0,0)$ to a plane is $(1,2,3)$, then equation of the plane is

If a line L makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with Y -axis and Z -axis respectively, then the angle between L and another line having direction ratios $1,1,1$ is