Question

Find the equation of the sphere which passes through the points (1,–3, 4), (1, –5, 2), (1, –3, 0) and has its centre on the plane x + y + z = 0.

Solution

Correct option is

 

Let the equation of the sphere be

                         …(1)

(1,–3, 4) is on the sphere.

 

                            …(2)

(1,–5, 2) is on the sphere.

                           …(3)

(1, –3, 0) is on the sphere. 

                                      …(4)

The centre (–u, –v, –w) of the sphere lies on the plane x + y + z = 0.

                                     …(5)

(2) – (3)                                                  …(6)

(3) – (4)                                         …(7)

(6) + (7)      

 

Putting the values of v and in (5), we get

       

Putting the values of u, v and w in (2), we get

 

 

 

 

This is the equation of the sphere.

SIMILAR QUESTIONS

Q1

Find the radius of the circular section of the sphere  by the plane                   

Q2

Find the centre and radius of the circle in which the plane  intersects the sphere 

Q3

Find the equation of the sphere passing through the points (0, 0, 0), (–1, 2, 0), (0, 1, –1) and (1, 2, 5).

Q4

Chord AB is a diameter of the sphere  with coordinates of A as (3, 2, –2). Find the coordinates of B.

Q5

A plane passes through a fixed point (­a, b, c) and cuts axes in A, B, C. Find the locus of the centre of the sphere OABC

Q6

Obtain the equation of the sphere described on the join of the points (2, –3, 4) and B(–5, 6, –7) as a diameter.

Q7

Obtain the equation of the sphere where the points (1, 0, 1) and (5, 4, 5) are the extremities of a diameter. Deduce the equation in Cartesian form. Also find the radius and the centre of the sphere.