3d planes practice · Mixed difficulty

Which one of the following planes contains the $z$-axis?

If the foot of the perpendicular drawn from $(1,9,7)$ to the line passing through the point $(3,2,1)$ and parallel to the planes $x+2 y+z=0$ and $3 y-z=3$ is $(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma$ is equal to

The sum of the intercepts on the coordinate axes of the plane passing through the point $(-2,-2,2)$ and containing the line joining the points $(1,-1,2)$ and $(1,1,1)$ is

If the line $\frac{{x - 1}}{2} = \frac{{y + 1}}{3} = \frac{{z - 2}}{4}$ meets the plane, $x + 2y + 3z = 15$ at a point $P$, then the distance $P$ from the origin is

The distance of the point (1, 1, 9) from the point of intersection of the line $\frac{{x - 3}}{1} = \frac{{y - 4}}{2} = \frac{{z - 5}}{2}$ and the plane $\,\,x + y + z = 17$ is

The locus of a point which moves in such a way that its distance from the line $\frac{x}{1}=\frac{y}{1}=\frac{z}{-1}$ is twice the distance from the plane $x+y+z=0$ is

Equation of plane passing through the points $(2,2,1),(9,3,6)$ and perpendicular to the plane $2 x+6 y+6 z-1=0$, is

If the axes are rectangular and ${P}$ is the point $(2,3,-1)$, then the equation of the plane through ${P}$ at right angles to $O P$, is-

A plane passes through the point $(1,1,1)$. If $b, c, a$ are the direction ratios of a normal to the plane where $a, b$, $c(a < b < c)$ are the prime factors of 2001, then the equation of the plane $1 \mathrm{l}$ is

The foot of perpendicular form the origin $\mathrm{O}$ to $\mathrm{a}$ plane $\mathrm{P}$ which meets the co-ordinate axes at the points $A, B, C$ is $(2, a, 4), a \in N$. If the volume of the tetrahedron OABC is 144 unit $^{3}$, then which of the following points is NOT on $P$?

The distance of the point $(1,-5,9)$ from the plane $x-y+z=5$ measured along the line $x=y=z$ is

If the shortest distance between the lines
$\frac{x-1}{\alpha}=\frac{y+1}{-1}=\frac{z}{1},(\alpha \neq-1)$ and $x+y+z+1=0$
$=2 x-y+z+3$ is $\frac{1}{\sqrt{3}}$, then a value $\alpha$ is :

A variable plane at a distance of 1 unit from the origin cuts the axes at $A, B$ and $C$. If the centroid $D(x, y, z)$ of $\triangle A B C$ satisfies the relation $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=K$, then the value of $K$ is

The equation of the plane passing through the mid-point of the line points $(1,2,3)$ and $(3,4,5)$ and perpendicular to it is

The distance of the point $(1,3,-7)$ from the plane passing through the point $(1,-1,-1)$ having normal perpendicular to both the lines $\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-4}{3}$ and $\frac{x-2}{2}=\frac{y+1}{-1}=\frac{z+7}{-1}$, is

The value of ' $\lambda$ ' for which the lines $3 x+2 y+z-5=0=x+y-2 z-3$ and $2 x-y-\lambda z=0=7 x+10 y-8 z$ are perpendicular to each other is

Two systems of rectangular axes have the same origin. If a plane cuts them at distance $a, b, c$ and $a^{\prime}, b^{\prime}, c^{\prime}$ from the origin, then :

A plane $P$ contains the line $x + 2y + 3z + 1 = 0 = x - y - z - 6\,$ and is perpendicular to the plane $- 2x + y + z + 8 = 0$. Then which of the following points lies on $P$?

A plane makes intercepts $O A, O B$ and $O C$ whose measurements are $a, b$ and $c$ on the $O X, O Y$ and $O Z$ axes. The area of triangle $A B C$ is

The line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}$ intersects the curve $x y=c^2$, in $x y$-plane, if $c$ is equal to

A plane meets the coordinate axes at $A,B,C$ such that the centroid of the triangle $ABC$ is at ${G} (a, b, c).$ The equation of the plane ${ABC}$ is-

Through the point $\mathrm{P}(1,2,2)$ a plane is drawn at right angles to OP meets the axes in A, B, C. Find $\lambda$, if the area of the triangle $\mathrm{ABC}$ is $(240+\lambda) / 8$.

The vector equation of the plane through the point $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and perpendicular to the line of intersection of the plane $\mathbf{r} \cdot(3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=1$ and $\mathbf{r} \cdot(\hat{\mathbf{i}}+4 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})=2$, is

The position vectors of points $\mathbf{a}$ and $\mathbf{b}$ are $\hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ respectively. The equation of a plane is r. $(5 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-7 \hat{\mathbf{k}})+9=0$. The points a and $\mathbf{b}$

The equation of the plane passing through the origin and perpendicular to the line $x=2 y=3 z$ is

If the distance between planes, $4 x-2 y-4 z+1=0$ and $4 x-2 y-4 z+d=0$ is 7 , then $d$ is:

If a plane passes through the points $(-1, k, 0)$, $(2, k,-1),(1,1,2)$ and is parallel to the line $\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}$, then the value of $\frac{k^2+1}{(k-1)(k-2)}$ is

Let $P$ be the plane, passing through the point $(1,-1,-5)$ and perpendicular to the line joining the points $(4,1,-3)$ and $(2,4,3)$. Then the distance of $P$ from the point $(3,-2,2)$ is

If the foot of the perpendicular from the point $A(-1,4,3)$ on the plane $P: 2 x+m y+n z=4$, is $\left(-2, \frac{7}{2}, \frac{3}{2}\right)$, then the distance of the point $A$ from the plane $P$, measured parallel to a line with direction ratios $3,-1,-4$, is equal to

The distance of the plane $\vec{r} \cdot\left(\frac{2}{7} \hat{i}+\frac{3}{7} \hat{j}-\frac{6}{7} \hat{k}\right)=1$ from the origin is