Question

3, a) Find the equation of the sphere whose centre is the point(x,y,z)and radius equal to r.
b) Find the equation of the sphere through the circle
x²+y²+z² = 25, x+2y-z+2=0) and the point (1,1,1)
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Answer 1

general equation of sphere,

x

2

+y

2

+2gx+2fy+2hz+c=0

centre (−g,−f.−h) and radius =

g

2

+f

2

+h

2

−c

At z=0, x

2

+y

2

+2gx+2fy+c=0−−−(i)

x

2

+y

2

−4=0−−−(ii)

comparing (i) and (ii), g=0,f=0,c=−4

∴ general equation reduces to,

x

2

+y

2

+z

2

+2hz+4=0 centre (0,0,h), radius=

h

2

−c

from fig., P is perpendicular drawn from centre to plane and r is radius of sphere. ∴p

2

+9=r

2

9

4h

2

+9=h

2

+4

⇒h=±3

∴ equation of spheres are,

x

2

+y

2

+z

2

+6z−4=0 and x

2

+y

2

+z

2

+6z−4=0