3d geometry system practice · Mixed difficulty

A variable line through the point $\left( {\frac{1}{5},\frac{1}{5}} \right)$ cuts the coordinate axes in the points
$A$ and $B$. If the point $P$ divides $AB$ internally in the ratio 3:1, then the locus of $P$ is

The $x y$-plane divides the line joining the points $(-1,3,4)(2,-5,6)$

The number of straight lines that are equally inclined to three dimensional coordinate axes, is

A point P moves on the line $2x - 3y + 4 = 0$. If Q(1, 4) and R(3, -2) are fixed points, then the locus of the centroid of $\Delta PQR$ is a line

If the line joining $A(1,3,4)$ and $B$ is divided by the point $(-2,3,5)$ in the ratio $1: 3$, then $B$ is

If the three points (0, 1), $\left( {0,\; - 1} \right)$ and $\left( {x,\;0} \right)$ are vertices of an equilateral triangle, then the values of $x$ are

The shortest distance (S.D) between lines $\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}-\frac{y+7}{2}=\frac{z-6}{4}$ is

The equation of the plane through the points $(2,3,1)$ and $(4,-5,3)$ and parallel to $x$-axis is

The line $\frac{x-2}{3}=\frac{y-3}{2}=\frac{z-4}{-4}$ intervals the curve $x y=k^2, z=0 \mathrm{x}$ then $k=$

If projection of a line on $x, y$ and $z$ axis are 6,2 and 3 respectively, then direction cosines of the line are

The locus of a point which divides the line segment joining the point $\left( {0, - 1} \right)$ and a point on the parabola, ${x^2} = 4y,$ internally in the ratio 1 : 2, is

The equation of the plane that passes through $(2,-3,1)$ and is perpendicular to the line joining the points $(3$, $4,-1)$ and $(2,-1,5)$ is

The equation of the plane through $(2,3,4)$ and parallel to the plane $x+2 y+4 z=5$ is

The ratio in which $Z X$-plane divides the line segment $A B$ joining the points $A(4,2,3)$ and $B(-2,4,5)$ is equal to

The distance between the planes ${r} \cdot(2 \hat{i}-\hat{j}+3 \hat{k})=4$ and ${r} \cdot(6 \hat{i}-3 \hat{j}+9 \hat{k})+13=0$ is

If $A B C D$ is a tetrahedron such that each $\triangle A B C, \triangle A B D$ and $\triangle A C D$ has a right angle at $A$. If $\operatorname{ar}(\triangle A B C)=k_1$, $\operatorname{ar}(\triangle A B D)=k_2, \operatorname{ar}(\triangle B C D)=k_3$, then $\operatorname{ar}(\triangle A C D)$ is

The ratio in which the line joining $(1,2,3)$ and $(-3,4,-5)$ is divided by $X Y$-plane, is

In a three dimensional co-ordinate system, $P, Q$ and $R$ are images of a point $A(a, b, c)$ in the $x y, y z$ and $z x$ planes, respectively. If $G$ is the centroid of triangle $P Q R$, then area of triangle $A O G$ is ( $O$ is the origin)

Chord ${AB}$ is a diameter of the sphere $|{r}-2 \hat{i}-\hat{j}+6 \hat{k}|$ $=\sqrt{18}$. If the coordinates of ${A}$ are $(3,2,-2)$, then the coordinates ${B}$ are

Equation of a line passing through the intersection of the lines $7 x-y+2=0$ and $x-3 y+6=0$ parallel to $x$-axis is

The centroid of a triangle is (2, 7) and two of its vertices are (4, 8) and $\left( { - 2,\;6} \right)$. The third vertex is

The centriod of the triangle $ABC$, where $A \equiv \left( {2,3} \right),B \equiv \left( {8,10} \right)$ and $C \equiv \left( {5,5} \right)$ is

If lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then the value of $k$ is

The points $(0,7,10),(-1,6,6)$ and $(-4,9,6)$ form

Three distinct points $A,\;B$ and $C$ given in the two dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point $\left( { - 1,\;0} \right)$ is equal to $\frac{1}{3}$. Then , the circumcentre of the triangle $ABC$ is at the point

The perpendicular distance of the point $P(6,7,8)$ from $X Y$-plane is ____.

The equation of the plane through the points $(2,-1,0)$, $(3,-4,5)$ and parallel to the line $2 x=3 y=4 z$ is

The perpendicular distance of the point $(6, 5, 8)$ from Y-axis is_________.

Four lines drawn from the vertices of any tetrahedron to the centroid of the opposite faces meet in a point whose distance from each vertex is $k$ times the distance from each vertex to the opposite face, where $k$ is

If the coordinates of the vertices of a $\Delta A B C$ are $A(-1,3,2), B(2,3,5)$ and $C(3,5,-2)$, then $\angle A$ is equal to