List of top Mathematics Questions on Distance of a Point from a Plane

The square of the distance of the point $\left( \frac{15}{7}, \frac{32}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\mathbf{i} + 4\mathbf{j} + 7\mathbf{k}$ is:

An equation of a plane parallel to the plane $x-2y+2z-5=0$ and at a unit distance from the origin is?

The shortest distance between the lines $\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}$ and $\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0}$ is

Let the equation of the plane P containing the line $x+10=\frac{8-y}{2}=z$ be $ax + by + 3z = 2(a + b)$ and the distance of the plane P from the point (1, 27, 7) be c. Then a²+b²+c² is equal to ____________.

Let $\lambda_1, \lambda_2$ be the values of $\lambda$ for which the points $(1, \lambda, \frac{1}{2})$ and $(-2, 0, 1)$ are at equal distance from the plane $2x + 3y - 6z + 7 = 0$. If $\lambda_1 > \lambda_2$, then the distance of the point $(1 - \lambda_2, \lambda_2, \lambda_1)$ from the line $\frac{x-5}{1} = \frac{y-1}{2} = \frac{z+7}{2}$ is:

Let P be the plane passing through the line $\frac{x-1}{1}=\frac{y-2}{-3}=\frac{z+5}{7}$ and the point (2, 4, –3). If the image of the point (-1, 3, 4) in the plane P is (α, β, γ) then α + β + γ is equal to

For $a, b \in \mathbb{Z}$ and $|a - b| \leq 10$, let the angle between the plane $P: ax + y - z = b$ and the line $L: x - 1 = a - y = z + 1$ be $\cos^{-1}\left(\frac{1}{3}\right)$. If the distance of the point $(6, -6, 4)$ from the plane $P$ is $3\sqrt{6}$, then $a^4 + b^2$ is equal to:

Let P₁ be the plane 3x-y-7z = 11 and P₂ be the plane passing through the points (2,-1,0), (2,0,-1), and (5,1,1). If the foot of the perpendicular drawn from the point (7,4,-1) on the line of intersection of the planes P₁ and P₂ is (α, β, γ), then a + ẞ+ y is equal to

Let P be the point of intersection of the line $\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}$ and the plane x + y + z = 2. If the distance of the point P from the plane 3x – 4y + 12z = 32 is q, then q and 2q are the roots of the equation

Let the foot of perpendicular of the point P(3, -2, -9) on the plane passing through the points (-1, -2, -3), (9, 3, 4), (9, -2, 1) be Q(α, β, γ). Then the distance of Q from the origin is

Let S be the set of all values of λ, for which the shortest distance between the lines x-λ/0 =y-3/4 = z+6/1 and x+λ/3 = y/-4 = z-6/0 is 13. Then 8 | $ \sum_{ λ∈S} λ| is $

Let the plane P contain the line 2x+y-z-3=0=5x-3y+4z+9 and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then the distance of the point
A(8, -1, -19) from the plane P measured parallel to the line $\frac{x}{-3}=\frac{y-5}{4}=\frac{2-z}{-12}$ is equal to ___________.

Let the plane x+3y–2z+6=0 meet the co-ordinate axes at the points A,B,C. If the orthocentre of the triangle ABC is$ (α,β,\frac{6 }{7} ), $ then 98(α + β)² is equal to___.

The plane through the intersection of the planes $x + y + z = 1$ and $2x + 3y - z + 4 = 0$ and parallel to y-axis also passes through the point :

The plane which bisects the line segment joining the points (-3,-3,4) and (3,7,6) at right angles, passes through which one of the following points ?

The foot of the perpendicular from the point $ (7, 14, 5) $ to the plane $ 2x + 4y - z = 7 $ is: