Advertisements
Advertisements
प्रश्न
Solve the following :
Find the distance of the point (13, 13, – 13) from the plane 3x + 4y – 12z = 0.
Advertisements
उत्तर
The distance of the point (x1, y1, z1) from the plane ax + by + cz + d = 0 is `|(ax_1 + by_1 + cz_1 + d)/sqrt(a^2 + b^2 + c^2)|`
∴ the distance of the point (1, 1, – 1) from the plane 3x + 4y – 12z = 0 is `|(3(1) + 4(1 - 12(-1)))/sqrt(3^2 + 4^2 + (-12)^2)|`
= `|(3 + 4 + 12)/sqrt(9 + 16 + 144)|`
= `(19)/sqrt(169)`
= `(19)/(13)`
= 19units.
APPEARS IN
संबंधित प्रश्न
Show that the points (1, 1, 1) and (-3, 0, 1) are equidistant from the plane `bar r (3bari+4barj-12bark)+13=0`
Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, −3), B(−2, −3, 5) and C(5, 3, −3).
Find the distance of a point (2, 5, −3) from the plane `vec r.(6hati-3hatj+2 hatk)=4`
In the given cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(0, 0, 0) 3x – 4y + 12 z = 3
In the given cases, find the distance of each of the given points from the corresponding given plane
Point Plane
(3, – 2, 1) 2x – y + 2z + 3 = 0
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
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 \[2 \hat{i} - \hat{j} - 4 \hat{k}\] from the plane \[\vec{r} \cdot \left( 3 \hat{i} - 4 \hat{j} + 12 \hat{k} \right) - 9 = 0 .\]
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, 3) from the plane x − y + z = 5 measured parallel to the line whose direction cosines are proportional to 2, 3, −6.
Find an equation for the set of all points that are equidistant from the planes 3x − 4y + 12z = 6 and 4x + 3z = 7.
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.
Find the equation of the plane mid-parallel to the planes 2x − 2y + z + 3 = 0 and 2x − 2y + z + 9 = 0.
Find the distance between the planes \[\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + 7 = 0 \text{ and } \vec{r} \cdot \left( 2 \hat{i} + 4 \hat{j} + 6 \hat{k} \right) + 7 = 0 .\]
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
Write the coordinates of the point which is the reflection of the point (α, β, γ) in the XZ-plane.
Find the distance of the point `4hat"i" - 3hat"j" + hat"k"` from the plane `bar"r".(2hat"i" + 3hat"j" - 6hat"k")` = 21.
The perpendicular distance of the origin from the plane x − 3y + 4z = 6 is ______
The equations of planes parallel to the plane x + 2y + 2z + 8 = 0, which are at a distance of 2 units from the point (1, 1, 2) are ________.
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`
Find the coordinates of the point where the line through (3, – 4, – 5) and (2, –3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0)
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
Find the distance of a point (2, 4, –1) from the line `(x + 5)/1 = (y + 3)/4 = (z - 6)/(-9)`
The distance of the plane `vec"r" *(2/7hat"i" + 3/4hat"j" - 6/7hat"k")` = 1 from the origin is ______.
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
A metro train starts from rest and in 5 s achieves 108 km/h. After that it moves with constant velocity and comes to rest after travelling 45 m with uniform retardation. If total distance travelled is 395 m, find total time of travelling.
`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 coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are
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.
Find the distance of the point (1, –2, 0) from the point of the line `vecr = 4hati + 2hatj + 7hatk + λ(3hati + 4hatj + 2hatk)` and the point `vecr.(hati - hatj + hatk)` = 10.
The acute angle between the line `vecr = (hati + 2hatj + hatk) + λ(hati + hatj + hatk)` and the plane `vecr xx (2hati - hatj + hatk)` is ______.
Find the coordinates of points on line `x/1 = (y - 1)/2 = (z + 1)/2` which are at a distance of `sqrt(11)` units from origin.
Let the foot of the perpendicular from the point (1, 2, 4) on the line `(x + 2)/4 = (y - 1)/2 = (z + 1)/3` be P.
Then the distance of P from the plane 3x + 4y + 12z + 23 = 0
