Sphere
Sphere
Equation of a Sphere when radius is given and center is at origin
- The equation of a sphere with radius r and centre at (0,0,0) is given by x2+y2+z2=r2.
Equation of a Sphere when radius and centre are given
- The equation of a sphere with radius r and centre at (a,b,c) is given by: (x−a)2+(y−b)2+(z−c)2=r2
which can also be represented as
x2+y2+z2−2ax−2by−2cz+(a2+b2+c2−r2)=0General form of the equation of a Sphere
- The general form of the equation of a sphere is given by:
x2+y2+z2+2ux+2vy+2wz+d=0 where u=−a, v=−b, w=−c and d=α2+b2+c2−r2
Equation of a Sphere when extreme points of the diameter are given
- The equation of a sphere with (x1,y1,z1) and (x2,y2,z2) at the extremes of a diameter is given by:
Equation of a Sphere passing through four non-coplanar points
- The equation of a sphere passing through four non-coplanar points is given by:
Condition for a plane to touch a Sphere
- The condition that the plane lx+my+nz=p touches the sphere x2+y2+z2+2ux+2vy+2wz+d=0 is given by:
Tangent Plane to a Sphere
- The equation of the tangent plane to the sphere x2+y2+z2+2ux+2vy+2wz+d=0 at point (x1,y1,z1) is given by:
Plane of Contact
- The locus of the point of contact of tangent plane to the sphere x2+y2+z2+2ux+2vy+2wz+d=0 which passes through (α,β,γ) is a circle which lies in the plane given by:
This plane is known as the plane of contact or contact plane.
General equation of Sphere passing through the circle created by the interesection of another Sphere and a plane
- The general equation of any sphere passing through the circle created by the intersection of the sphere x2+y2+z2+2ux+2vy+2wz+d=0 and the plane Ax+By+Cz+D=0 is given by:
x2+y2+z2+2ux+2vy+2wz+d+k(Ax+By+Cz+D)=0.
Great Circle
- The plane section of a sphere is a circle and if the plane passes through the center of the sphere, then the section is called the great circle.
Angle between two Spheres
- The angle between two spheres is calculated as: cosθ=r21+r22−d22r1r2, where d is the distance between the spheres. It follows that for orthogonality, d2=r21+r22.
Condition for orthogonal intersection of two Spheres
- The condition that two spheres x2+y2+z2+2u1x+2v1y+2w1z+d1=0 and x2+y2+z2+2u2x+2v2y+2w2z+d2=0 interesects orthogonally is given by:
Length of tangent from a point to a Sphere
- The length of the tangent drawn from P(x1,y1,z1) to the sphere x2+y2+z2+2ux+2vy+2wz=0 is given by:
PYQs
Sphere
1) Find the equation of the sphere in xyz−plane passing through the points (0,0,0), (0,1,−1), (−1,2,0), (1,2,3).
[2018, 12M]
2) A plane passes through a fixed point (a,b,c) and cuts the axes at the points A, B, C respectively. Find the locus of the center of the sphere which passes through the origin O and A, B, C.
[2017, 15M]
3) Show that the plane 2x−2y+z+12=0 touches the sphere x2+y2+z2−2x−4y+2z−3=0. Find the point of contact.
[2017, 10M]
4) Find the equation of the sphere which passes though the circle x2+y2=4; z=0 and is cut by the plane x+2y+2z=0 in a circle of radius 3.
[2016, 10M]
5) Find the positive value of a for which the plane ax−2y+z+12=0 touches the sphere x2+y2+z2−2x−4y+2z−3=0. Also find the point of contact.
[2015, 10M]
6) Find the co-ordinates of the points on the sphere x2+y2+z2−4x+2y=4 the tangent planes at which are parallel to the plane 2x−y+2z=1.
[2014, 10M]
7) A sphere S has points (0,1,0),(3,−5,2) at opposite ends of a diameter. Find the equation of the sphere having the intersection of the sphere S with the plane 5x−2y+4z+7=0 as a great circle.
[2013, 10M]
8) Show that three mutually perpendicular tangent lines can be drawn to the sphere x2+y2+z2=r2 from any point on the sphere 2(x2+y2+z2)=3r2.
[2013, 15M]
9) Find the points on the sphere x2+y2+z2=4 that are closest to and farthest from the point (3,1,−1).
[2011, 20M]
10) Show that the plane x+y−2z=3 cuts the sphere x2+y2+z2−x+y=2 in a circle of radius 1 and find the equation of the sphere which has this circle as a great circle.
[2011, 12M]
11) Show that the equation of the sphere which touches the sphere 4(x2+y2+z2)+10x−25y−2z=0 at the point (1,2,−2) and passes through the point (−1,0,0) is x2+y2+z2+2x−6y+1=0.
[2010, 10M]
12) Show that every sphere through the circle x2+y2−2ax+r2=0,z=0 cuts orthogonally every sphere through the circle x2+z2=r2,y=0.
[2010, 20M]
13) Find the equation of the sphere having its centre on the plane 4x−5y−z=3, and passing through the circle
x2+y2+z2−12x−3y+4z+8=0; 3x+4y−5z+3=0[2009, 12M]
14) Find the equations (in symmetric form) of the tangent line to the sphere x2+y2+z2+5x−7y+2z−8=0, 3x−2y+4z+3=0 at the point (−3,5,4).
[2008, 12M]
15) A sphere S has points (0,1,0), (3,−5,2) at opposite ends of a diameter. Find the equation of the sphere having the intersection of the sphere S with the plane 5x−2y+4z+7=0 as a great circle.
[2008, 20M]
16) Find the equation of the sphere inscribed in the tetraahedron whosw faces are x=0, y=0, S$z=0and3x+3y+6z=6$$.
[2007, 12M]
17) Show that the spheres x2+y2+z2−x+z−2=0 and 3x2+3y2−8x−10y+8z+14=0 cut orthogonally. Find the center and radius of their common circle.
[2007, 15M]
18) Show that the locus of the centers of sphere of a co-axial system is a straight line.
[2005, 15M]
19) Find the equations of the tangent planes to the sphere x2+y2+z2−4x+2y−6z+5=0, which are parallel to the plane 2x+y−z=4.
[2004, 12M]
20) A sphere of constant radius r passes through the origin O and cuts the co-ordinate axes at A, B and C. Find the locus of the foot of the perpendicular from O to the plane ABC.
[2003, 15M]
21) Find the coordinates of the centre of the sphere inscribed in the tetrahedron formed by the planes
x=0, y=0, z=0 and x+y+z=a.
[2002, 12M]