Question

find equation of sphere through the circle x² +y² +z²= 1 , 2x+ 4y +5z -6=0 and touch z=0​

Answer 1

Answer:

Equation to the sphere passes through the circle x 2 +y 2+z 2

=5,x+2y+3z=3 is given by,

x 2 +y 2 +z 2−5+λ(x+2y+3z−3)=0

⇒x 2+y 2+z 2 +λx+2λy+3λz−(3λ+5)=0

Given plane 4x+3y−15=0 touches this sphere,∣∣∣l∣54( 2−λ )+3(−λ)−15

∣∣∣∣∣

= ( 2λ ) 2 +λ 2+( 23λ +3λ+5) 2

⇒5λ 2−6λ−8=0⇒λ=2,−4/5

Hence, the sphere is x 2+y 2 +z 2+2x+4y+6z−11=0

There will be one more sphere corresponding to another value of λ.