मराठी

Find the equation of a plane which is at a distance 33 units from origin and the normal to which is equally inclined to coordinate axis - Mathematics

Advertisements
Advertisements

प्रश्न

Find the equation of a plane which is at a distance `3sqrt(3)` units from origin and the normal to which is equally inclined to coordinate axis

बेरीज
Advertisements

उत्तर

If α, β, and γ are the angles made by the line segment with the coordinate axis.
 
cosα, cosβ and cosγ are called as the direction cosines.
 
Let the required equation of the plane be `vecr * hatn` = p, where p = `3sqrt(3)`.
 
Let `vecn = (cosalpha)hati + (cosalpha)hatj + (cosalpha)hatk`, where α is acute.
 
Then, `cos^2alpha + cos^2alpha + cos^2alpha` = 1
 
⇒ `3cos^2alpha` = 1
 
⇒ `cos^2alpha = 1/3`
 
⇒ `cos alpha = 1/sqrt(3)`
 
∴ The required equation is `vecr * (1/sqrt(3)hati + 1/sqrt(3)hatj + 1/sqrt(3)hatk) = 3sqrt(3)`
 
Hence, `vecr * (hati + hatj + hatk)` = 9
 
⇒ `(xhati + yhatj + zhatk) * (hati + hatj + hatk)` = 9
 
⇒ x + y + z = 9. 
shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 12: Introduction to Three Dimensional Geometry - Exercise [पृष्ठ २३५]

APPEARS IN

एनसीईआरटी एक्झांप्लर Mathematics [English] Class 11
पाठ 12 Introduction to Three Dimensional Geometry
Exercise | Q 8 | पृष्ठ २३५

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

Find the distance between the following pairs of points:

(–3, 7, 2) and (2, 4, –1)


Find the distance between the following pairs of points:

(2, –1, 3) and (–2, 1, 3)


Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.


Verify the following:

(0, 7, –10), (1, 6, –6) and (4, 9, –6) are the vertices of an isosceles triangle.


Verify the following:

(0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of a right angled triangle.


Verify the following:

(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are the vertices of a parallelogram.


Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (–4, 0, 0) is equal to 10.


Find the distance between the following pairs of points: 

P(1, –1, 0) and Q(2, 1, 2)


Find the distance between the points P and Q having coordinates (–2, 3, 1) and (2, 1, 2).


Using distance formula prove that the following points are collinear: 

P(0, 7, –7), Q(1, 4, –5) and R(–1, 10, –9)


Show that the points (0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of an isosceles right-angled triangle. 


Prove that the tetrahedron with vertices at the points O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a regular one.


Show that the points (3, 2, 2), (–1, 4, 2), (0, 5, 6), (2, 1, 2) lie on a sphere whose centre is (1, 3, 4). Find also its radius.


Find the centroid of a triangle, mid-points of whose sides are (1, 2, –3), (3, 0, 1) and (–1, 1, –4). 


The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of and are (3, –5, 7) and (–1, 7, –6) respectively, find the coordinates of the point C.


If the distance between the points P(a, 2, 1) and Q (1, −1, 1) is 5 units, find the value of a


Write the coordinates of third vertex of a triangle having centroid at the origin and two vertices at (3, −5, 7) and (3, 0, 1). 


Find the distance of the point (– 2, 4, – 5) from the line `(x + 3)/3 = (y - 4)/5 = (z + 8)/6`


Find the distance of the point (–1, –5, – 10) from the point of intersection of the line `vecr = 2hati - hatj + 2hatk + lambda(3hati + 4hatj + 2hatk)` and the plane `vecr * (hati - hatj + hatk)` = 5.


The distance of a point P(a, b, c) from x-axis is ______.


Find the angle between the lines `vecr = 3hati - 2hatj + 6hatk + lambda(2hati + hatj + 2hatk)` and `vecr = (2hatj - 5hatk) + mu(6hati + 3hatj + 2hatk)`


Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(– 4, 4, 4).


Find the distance of a point (2, 4, –1) from the line `(x + 5)/1 = (y + 3)/4 = (z - 6)/(-9)`


Distance of the point (α, β, γ) from y-axis is ______.


The distance of the plane `vecr * (2/4 hati + 3/7 hatj - 6/7hatk)` = 1 from the origin is ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×