Find the equation of the sphere which has centre at the origin and touches the line `2(x+1)=2-y=z+3.`

Explanation

The equation of the given line is `2(x+1)=2-y=z+3`.
or `" "(x+1)/(1//2) = (y-2)/(-1)=(z+3)/(1) = lamda` (say)
Let the point of contact be `N(-1+(1)/(2)lamda, 2-lambda, -3+lamda)`
Center of sphere is origin O.
ON is perpendicular to the given line. Then
`" "(1)/(2)(-1+(1)/(2)lamda)+(-1)(2-lamda)+1(-3+lamda)=0`
`therefore " "lamda=(22)/(9)`
Hence, the point of contact is `((2)/(9),-(4)/(9), -(5)/(9))`
Thus, equation of the sphere is `x^(2)+y^(2)+z^(2)=(4)/(81)+(16)/(81)+(25)/(81)`
or `" "9(x^(2)+y^(2)+z^(2))=5`