Advertisements
Advertisements
प्रश्न
If a plane passes through the point (1, 1, 1) and is perpendicular to the line \[\frac{x - 1}{3} = \frac{y - 1}{0} = \frac{z - 1}{4}\] then its perpendicular distance from the origin is ______.
विकल्प
`3/4`
`4/3`
`7/5`
1
Advertisements
उत्तर
If a plane passes through the point (1, 1, 1) and is perpendicular to the line \[\frac{x - 1}{3} = \frac{y - 1}{0} = \frac{z - 1}{4}\] then its perpendicular distance from the origin is `underlinebb(7/5)`.
Explanation:
Since the plane is perpendicular to the given line, its direction ratios are proportional to 3, 0, 4
So, the required equation of the plane is of the form
\[3x + 0y + 4z + d = 0 ... \left( 1 \right), \text{where d is a constant}.\]
\[\text{Since this plane passes through} (1, 1, 1),\]
\[3 + 0 + 4 + d = 0\]
\[\Rightarrow d = - 7\]
\[\text{Substituting this in (1), we get}\]
\[3x + 0y + 4z - 7 = 0 ... \left( 2 \right)\]
\[\text{Perpendicular distance of (2) from the origin}\]
\[= \frac{\left|3\left(0\right) + 0 + 4 \left(0\right) - 7 \right|}{\sqrt{3^2 + 0^2 + 4^2}}\]
\[= \frac{\left| 0 + 0 - 7 \right|}{\sqrt{25}}\]
\[= \frac{7}{5} \text{units}\]
संबंधित प्रश्न
Find the equation of the planes parallel to the plane x + 2y+ 2z + 8 =0 which are at the distance of 2 units from the point (1,1, 2)
In the given cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(2, 3, – 5) x + 2y – 2z = 9
Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
(A) 2 units
(B) 4 units
(C) 8 units
(D)`2/sqrt29 "units"`
Show that the points (1, –1, 3) and (3, 4, 3) are equidistant from the plane 5x + 2y – 7z + 8 = 0
Write the equation of a plane which is at a distance of \[5\sqrt{3}\] units from origin and the normal to which is equally inclined to coordinate axes.
Find the distance of the point (2, 3, −5) from the plane x + 2y − 2z − 9 = 0.
Show that the points (1, 1, 1) and (−3, 0, 1) are equidistant from the plane 3x + 4y − 12z + 13 = 0.
Find the distance of the point (1, -2, 4) from plane passing throuhg the point (1, 2, 2) and perpendicular of the planes x - y + 2z = 3 and 2x - 2y + z + 12 = 0
Find the distance between the parallel planes 2x − y + 3z − 4 = 0 and 6x − 3y + 9z + 13 = 0.
The distance between the planes 2x + 2y − z + 2 = 0 and 4x + 4y − 2z + 5 = 0 is
The image of the point (1, 3, 4) in the plane 2x − y + z + 3 = 0 is
The distance of the line \[\vec{r} = 2 \hat{i} - 2 \hat{j} + 3 \hat{k} + \lambda\left( \hat{i} - \hat{j}+ 4 \hat{k} \right)\] from the plane \[\vec{r} \cdot \left( \hat{i} + 5 \hat{j} + \hat{k} \right) = 5\] is
Find the distance of the point `4hat"i" - 3hat"j" + hat"k"` from the plane `bar"r".(2hat"i" + 3hat"j" - 6hat"k")` = 21.
Solve the following:
Find the distance of the point `3hat"i" + 3hat"j" + hat"k"` from the plane `bar"r".(2hat"i" + 3hat"j" + 6hat"k")` = 21.
Solve the following :
Find the distance of the point (13, 13, – 13) from the plane 3x + 4y – 12z = 0.
The equation of the plane passing through (3, 1, 2) and making equal intercepts on the coordinate axes is _______.
If the foot of perpendicular drawn from the origin to the plane is (3, 2, 1), then the equation of plane is ____________.
Find the distance of the point (– 2, 4, – 5) from the line `(x + 3)/3 = (y - 4)/5 = (z + 8)/6`
A plane meets the co-ordinates axis in A, B, C such that the centroid of the ∆ABC is the point (α, β, γ). Show that the equation of the plane is `x/alpha + y/beta + z/ϒ` = 3
The distance of the plane `vec"r" *(2/7hat"i" + 3/4hat"j" - 6/7hat"k")` = 1 from the origin is ______.
Find the equation of the plane passing through the point (1, 1, 1) and is perpendicular to the line `("x" - 1)/3 = ("y" - 2)/0 = ("z" - 3)/4`. Also, find the distance of this plane from the origin.
Find the foot of the perpendicular from the point (1, 2, 0) upon the plane x – 3y + 2z = 9. Hence, find the distance of the point (1, 2, 0) from the given plane.
Which one of the following statements is correct for a moving body?
S and S are the focii of the ellipse `x^2/a^2 + y^2/b^2 - 1` whose one of the ends of the minor axis is the point B If ∠SBS' = 90°, then the eccentricity of the ellipse is
`phi` is the angle of the incline when a block of mass m just starts slipping down. The distance covered by the block if thrown up the incline with an initial speed u0 is
The equations of motion of a rocket are:
x = 2t,y = –4t, z = 4t, where the time t is given in seconds, and the coordinates of a ‘moving point in km. What is the path of the rocket? At what distances will the rocket be from the starting point O(0, 0, 0) and from the following line in 10 seconds? `vecr = 20hati - 10hatj + 40hatk + μ(10hati - 20hatj + 10hatk)`
Find the distance of the point (2, 3, 4) measured along the line `(x - 4)/3 = (y + 5)/6 = (z + 1)/2` from the plane 3x + 2y + 2z + 5 = 0.
If the distance of the point (1, 1, 1) from the plane x – y + z + λ = 0 is `5/sqrt(3)`, find the value(s) of λ.
The distance of the point `2hati + hatj - hatk` from the plane `vecr.(hati - 2hatj + 4hatk)` = 9 will be ______.
Find the equations of the planes parallel to the plane x – 2y + 2z – 4 = 0 which is a unit distance from the point (1, 2, 3).
In the figure given below, if the coordinates of the point P are (a, b, c), then what are the perpendicular distances of P from XY, YZ and ZX planes respectively?
A plane passes through (1,-2,1) and is perpendicular to the planes 2x - 2y + z = 0 and x - y + 2z = 4. The distance of the point (1, 2, 2) from this plane is______ units.
