Let and be the endpoints of a diameter of a sphere.

MATHEMATICS · Multi-part · IB Diploma Programme · Mathematics: Analysis and Approaches · Geometry and Trigonometry · Geometry and Trigonometry

Problem

Let A(x1,y1,z1)A(x_1,y_1,z_1) and B(x2,y2,z2)B(x_2,y_2,z_2) be the endpoints of a diameter of a sphere. (i) Explain why the midpoint OO of ABAB is the centre of the sphere. (ii) Show that the length of the radius is 12(x2−x1)2+(y2−y1)2+(z2−z1)2\dfrac12\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}. (iii) Hence derive the general equation of the sphere whose diameter is ABAB.

Related theory

Browse the problem library