List of top Mathematics Questions on Plane

Let R be the relation "is congruent to" on the set of all triangles in a plane. Is R:

Let the acute angle bisector of the two planes \( x - 2y - 2z + 1 = 0 \) and \( 2x - 3y - 6z + 1 = 0 \) be the plane \( P \). Then which of the following points lies on \( P \)?

Let the foot of perpendicular from a point \( P(1,2,-1) \) to the straight line \( L : \frac{x}{1} = \frac{y}{0} = \frac{z}{-1} \) be \( N \). Let a line be drawn from \( P \) parallel to the plane \( x + y + 2z = 0 \) which meets \( L \) at point \( Q \). If \( \alpha \) is the acute angle between the lines \( PN \) and \( PQ \), then \( \cos \alpha \) is equal to:

If the plane \[ x - y + z + 4 = 0 \] divides the line joining the points \[ P(2,3,-1) \quad {and} \quad Q(1,4,-2) \] in the ratio \( l:m \), then \( l + m \) is:

The distance between the two planes $2x + 3y + 4z = 4$ and $4x + 6y + 8z = 12$ is:

Let \( P(x_1, y_1, z_1) \) be the foot of the perpendicular drawn from the point \[ Q(2, -2, 1) \] to the plane \[ x - 2y + z = 1. \] If \( d \) is the perpendicular distance from the point \( Q \) to the plane and \[ I = x_1 + y_1 + z_1, \] then \( I + 3d^2 \) is:

Let P be the plane √3x+2y+3z =16 and let and let S = {\(\alpha \hat{i}+\beta \hat{j}+\gamma\hat{k}:\alpha^2+\beta^2+\gamma^2=1\) and the distance of (α, β, γ) from the plane P is \(\frac{7}{2}\) }. Let u, v, and w be three distinct vectors in s such that |\(\vec{u}-\vec{v}\)| = |\(\vec{v}-\vec{w}\)| = |\(\vec{w}-\vec{u}\)|. Let V be the volume of the parallelepiped determined by vectors \(\vec{u},\vec{v},\vec{w}\). Then the value of \(\frac{80}{\sqrt3}\)V is

Let P be the plane √3x+2y+3z =16 and let and let S = {\(\alpha \hat{i}+\beta \hat{j}+\gamma\hat{k}:\alpha^2+\beta^2+\gamma^2=1\) and the distance of (α, β, γ) from the plane P is \(\frac{7}{2}\) }. Let u, v, and w be three distinct vectors in s such that |\(\vec{u}-\vec{v}\)| = |\(\vec{v}-\vec{w}\)| = |\(\vec{w}-\vec{u}\)|. Let V be the volume of the parallelepiped determined by vectors \(\vec{u},\vec{v},\vec{w}\). Then the value of \(\frac{80}{\sqrt3}\)V is

The plane $ x - 2y + z = 0 $ is parallel to the line

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

The locus of points(x,y) in the plane satisfying sin²x + sin²y =1 consists of

A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\prime}$, 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

A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\prime}$, 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

If the equation of the plane passing through the point $(1,1,2)$ and perpendicular to the line $x-3 y+$ $2 z-1=0=4 x-y+z$ is $A t+ B y+ C z=1$, then $140( C - B + A )$ is equal to______

Let Q be the foot of perpendicular drawn from the point P(1, 2, 3) to the plane x + 2y + z = 14. If R is a point on the plane such that ∠PRQ = 60°, then the area of ΔPQR is equal to :

Let l₁ be the line in xy-plane with x and y intercepts ⅛ and \(\frac{1}{4√2} \) respectively and l₂ be the line in zx-plane with x and z intercepts -⅛ and \(−\frac{1}{6√3}\) respectively. If d is the shortest distance between the line l₁ and l₂, then d^–2 is equal to ______.

Let \(P1:\overrightarrow{r}.(2\^i+\^j−3\^k )=4\) be a plane. Let \(P_2\) be another plane which passes through points  \((2, - 3, 2)\), \((2, - 2, -3)\) and \((1, -4, 2)\). If the direction ratios of the line of intersection of \(P_1\) and \(P_2\) be \(16, α,β,\) then the value of \(α + β\) is equal to _____ .

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi}{4}$, if the equation $25 x^2 + 9y^2 = 225$ is transformed to $\alpha x^2 + \beta xy + \gamma y^2 = \delta$, then $(\alpha + \beta + \gamma - \sqrt{\delta})^2$ =