91Ó°ÊÓ

Find the equation of the sphere with (i) centre at \((1,-2,3)\) and radius 5 units. (ii) centre at \(\left(-\frac{1}{3}, \frac{2}{3}, \frac{1}{3}\right)\) and radius 1 unit. (iii) centre at \((1,2,3)\) and radius 4 units.

Find the condition that the plane \(l x+m y+n z=p\) may touch the sphere \(x^{2}+y^{2}\) \(+z^{2}+2 u x+2 v y+2 w z+d=0 .\)

Prove that the sphere circumscribing the tetrahedron whose faces are \(y+z=0\), \(z+x=0, x+y=0\) and \(x+y+z=1\) is \(x^{2}+y^{2}+z^{2}-3(x+y+z)=0 .\)

A point moves such that, the sum of the squares of its distances from the six faces of a cube is a constant. Prove that its locus is the sphere \(x^{2}+y^{2}+z^{2}=3\left(k^{2}-a^{2}\right)\).

Prove that the spheres \(x^{2}+y^{2}+z^{2}=100\) and \(x^{2}+y^{2}+z^{2}-12 x+4 y-6 z+40=0\) touch internally and find the point of contact.

Prove that the spheres \(x^{2}+y^{2}+z^{2}=25\) and \(x^{2}+y^{2}+z^{2}-24 x-40 y-18 z+\) \(225=0\) touch externally. Find the point of contact.

Find the equation of the sphere which touches the coordinate planes and whose centre lies in the first octant.

Find the equation of the sphere on the line joining the points: (i) \((4,-1,2)\) and \((2,3,6)\) as the extremities of a diameter (ii) \((2,-3,4)\) and \((-5,6,-7)\) as the extremities of a diameter