3d geometry system practice · Easy

If the vertices $P,\;Q,\;R\;$ of a $\Delta \;PQR$ are rational points, which of the following points of the $\Delta \;PQR$ is (are) always rational points?

(A rational point is a point both of whose coordinates are rational numbers)

A straight line which makes an angle of $60^{\circ}$ with each of $y$ and $z$-axis, inclines with $x$-axis at an angle

The equation of the line passing through the point with position vector $2 \hat{i}-3 \hat{j}-5 \hat{k}$ and perpendicular to the plane ${r} \cdot(6 \hat{i}-3 \hat{j}-5 \hat{k})+2=0$ is

The equation of straight line passing through the points $(a, b, c)$ and $(a-b, b-c, c-a)$, is

The coordinates of the centroid of a triangle having its circumcentre and orthocentre at $\left( {7/2,\;5/2} \right)$ and $\left( {2,1} \right)$ respectively, are

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

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

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

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

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

If the projection of a line segment on $x, y$ and $z$ axes are respectively 3,4 and 5 , then the length of the line segment is

If the origin is the centroid of a $\Delta A B C$ having vertices $A(a, 1,3), B(-2, b,-5)$ and $C(4,7, c)$, then

Given that ${A}(3,2,-4), {B}(5,4-6)$ and ${C}(9,8,-10)$ are collinear. Ratio in which $B$ divides $AC$ is $1:{m}$. The value of ${m}$ is

Vertices of a variable triangle are $(3,4),(5 \cos \theta, 5 \sin \theta)$ and $(5 \sin \theta,-5 \cos \theta)$, where $\theta \in R$. Locus of its orthocentre is

If the projections of a line on the axes are $6,2,3$, then its direction cosines are

The locus of a points which moves such that the sum of the squares of its distance from three vertices of the triangle is constant is a/an