English

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

Question

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

Sum
Advertisements

Solution

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
  Is there an error in this question or solution?
Chapter 12: Introduction to Three Dimensional Geometry - Exercise [Page 235]

APPEARS IN

NCERT Exemplar Mathematics [English] Class 11
Chapter 12 Introduction to Three Dimensional Geometry
Exercise | Q 8 | Page 235

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

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:

(–1, 3, –4) and (1, –3, 4)


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 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 which are equidistant from the points (1, 2, 3) and (3, 2, –1).


Find the distance between the following pairs of points: 

P(1, –1, 0) and Q(2, 1, 2)


Find the distance between the following pairs of point: 

A(3, 2, –1) and B(–1, –1, –1).


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:

A(4, –3, –1), B(5, –7, 6) and C(3, 1, –8)


Using distance formula prove that the following points are collinear: 

A(3, –5, 1), B(–1, 0, 8) and C(7, –10, –6)


Determine the points in xy-plan are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).


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


Show that the points A(1, 3, 4), B(–1, 6, 10), C(–7, 4, 7) and D(–5, 1, 1) are the vertices of a rhombus. 


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.


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 whose position vector is `(2hati + hatj - hatk)` from the plane `vecr * (hati - 2hatj + 4hatk)` = 9


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


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)`


Find the equation of the plane through the intersection of the planes `vecr * (hati + 3hatj) - 6` = 0 and `vecr * (3hati + hatj + 4hatk)` = 0, whose perpendicular distance from origin is unity.


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 ______.


If one of the diameters of the circle x2 + y2 – 2x – 6y + 6 = 0 is a chord of another circle 'C' whose center is at (2, 1), then its radius is ______.


The points A(5, –1, 1); B(7, –4, 7); C(1, –6, 10) and D(–1, –3, 4) are vertices of a ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×