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Sphere

December 8, 2016
Published By : Pratik Kataria
Categorised in:

Equations of sphere in different forms

(A) Centre and radius form

$(x – a)^2 + (y – b)^2 + (z-c)^2 = r^2$

(B) General form

$x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$


Centre: (-u, -v, -w)

Radius: $r=√{u^2 + v^2 + w^2 – d}$

(C)Intercept form: To find the equation of the sphere which cuts off intercepts a, b, c from ox, oy, oz axes:

$x^2 + y^2 + z^2 – ax – by – cz = 0$

(D) Diameter form: To find the equation of the sphere described on the join of two given points as diameter:

Using condition of perpendicularity,

$(x – x_1) (x – x_2) + (y – y_1) (y – y_2) + (z – z_1) (z – z_2) = 0$

Coordinates of Centre: ${x_1 + x_2}/2, {y_1 + y_2}/2 , {z_1 + z_2}/2$

(E) Four-point form

$|\table x^2 + y^2 + z^2, x,y,z,1;x^2_1 + y^2_1 + z^2_1,x_1,y_1,z_1,1;x^2_2+y^2_2 +z^2_2,x_2,y_2,z_2,1;x^2_3+y^2_3+z^2_3,x_3,y_3,z_3,1;x^2_4+y^2_4+z^2_4,x_4,y_4,z_4,1|$ = 0

Tangent plane of the sphere

$x x_1 + y y_1 + z z_1 + u (x + x_1) + v (y + y_1) + w (z + z_1) + d = 0$

Section of sphere by a plane

${x+u}/l = {y+v}/m = {z+w}/n$

Intersection of two spheres

$S_1 – S_2 = 2(u_1-u_2)x + 2(v_1 + v_2) y + 2 (w_1 + w_2) + d_1 – d_2 = 0$

Equations of circle

S = 0 (Equation of a sphere)

U = 0 (Equation of a plane)

Together represent a circle.


Also, Intersection of two spheres

$S_1 = 0$

$S_2 = 0$

Sphere through a circle

$x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d + λ(lx + my + nz – p) = 0$


S + λU = 0

$S_1 + λ S_2 = 0$

$(x^2 + y^2 + z^2) (1 + λ) + 2x (u_1 + λ u_2) + 2y (v_1 + λ v_2) + 2z (w_1 + λ w_2) + d_1 + λ d_2 = 0$

Orthogonal spheres

$C_1 C^2_2 = C_1 P^2 + C_2 P^2$

$2 u_1 u_2 + 2 v_1 v_2 + 2 w_1 w_2 = d_1 + d_2$

Length of the tangent

$AT^2 = x^2_1 + y^2_1 + z^2_1 + 2u_1 x_1 + 2 v_1 y_1 + 2 w_1 z_1 + d_1$

Radical plane

$2x (u_1 – u_2) + 2y (v_1 – v_2) + 2z (w_1 + w_2) + d – d_2 = 0$


In short, $S_1 – S_2 = 0$