Understand and use equations of spheres. equation of sphere: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 + (𝑧 − 𝑙)2 = 𝑟2

A sphere has centre at (5, −2, 3) and radius √41. What is the equation of the sphere?

A sphere has centre \( C(2, -3, 5) \) and passes through the point \( P(5, 1, 9) \). a) Determine the radius of the sphere. (1 mark) b) Hence, write down the equation of the sphere in the form \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \). (1 mark) c) Show that the point \( Q(4, -1, 2) \) lies inside the sphere. (1 mark)

A sphere passes through the points \(P(2, 1, 3)\), \(Q(4, 3, 1)\), \(R(6, 1, 3)\), and \(S(4, 1, 5)\). (a) Determine the coordinates of the centre of the sphere. (3 marks) (b) Hence, find the radius of the sphere, correct to two decimal places. (1 mark)

A sphere passes through the point \(P(5, 2, 7)\) and has centre \(C(-1, 4, 3)\). (a) Determine the radius of the sphere. (1 mark) (b) Write the equation of the sphere in the form \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\). (2 marks)

A satellite tracking station can detect signals from any spacecraft within a spherical detection zone. The tracking station is located at point T with coordinates (5, -3, 8) in a three-dimensional coordinate system where units are measured in thousands of kilometres. A spacecraft at point S with coordinates (9, 1, 2) is currently at the outer edge of the detection zone. (a) Determine the length of the radius of the detection zone. (1 mark) (b) Determine the equation of the sphere representing the detection zone. (1 mark) (c) A second spacecraft is located at point P with coordinates (7, -6, 11). Determine whether this spacecraft is inside, outside, or on the boundary of the detection zone. Show your working. (1 mark)

A sphere passes through the points \(P(5, 2, 1)\), \(Q(3, 4, 1)\), \(R(5, 0, 1)\), and \(S(5, 2, 7)\). (a) Determine the coordinates of the centre of the sphere. (2 marks) (b) Hence find the equation of the sphere in the form \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\). (1 mark)

A sphere \(S_1\) has centre \(C_1\) at \((2, -3, 5)\) and passes through the point \(P(4, 1, 3)\). (a) Determine the equation of sphere \(S_1\). (2 marks) (b) A second sphere \(S_2\) has equation \((x + 1)^2 + (y - 2)^2 + (z + 4)^2 = 64\). Determine the distance between the centres of the two spheres, correct to two decimal places. (2 marks)