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PyMaths Academy
PyMaths Academy

Sphere

We will cover following topics

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)=0

General 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:
(x−x1)(x−x2)+(y−y1)(y−y2)+(z−z1)(z−z2)=0

Equation of a Sphere passing through four non-coplanar points

  • The equation of a sphere passing through four non-coplanar points is given by:
|x2+y2+z2xyz1x21+y21+z21x1y1z11x22+y22+z22x2y2z21x23+y23+z23x3y3z31x24+y24+z24x4y4z41|=0

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:
(lu+mv+nw+p)2=(l2+m2+n2)(u2+v2+w2−d)

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:
xx1+yy1+zz1+u(x+x1)+v(y+y1)+w(z+z1)+d=0

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:
αx+βy+γz+u(x+α)+v(y+β)+w(z+γ)+d=0

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:
2(u1u2+v1v2+w1w2)=d1+d2

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:
√x21+y21+z21+2ux1+2vy1+2wz1+d

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]

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