Question
The number of integral values of $$\lambda $$ for which $${x^2} + {y^2} + \lambda x + \left( {1 - \lambda } \right)y + 5 = 0$$ is the equation of a circle whose radius cannot exceed 5, is :
A.
$$14$$
B.
$$18$$
C.
$$16$$
D.
none of these
Answer :
$$16$$
Solution :
$$\eqalign{
& {\text{Here radius}} = \sqrt {{{\left( {\frac{\lambda }{2}} \right)}^2} + {{\left( {\frac{{1 - \lambda }}{2}} \right)}^2} - 5} \leqslant 5 \cr
& \Rightarrow 2{\lambda ^2} - 2\lambda - 119 \leqslant 0 \cr
& \therefore \,\,\frac{{1 - \sqrt {239} }}{2} \leqslant \lambda \leqslant \frac{{1 + \sqrt {239} }}{2} \cr
& \Rightarrow - 7.2 \leqslant \lambda \leqslant 8.2\,\left( {{\text{nearly}}} \right)\,\,\,\,\,\,\,\therefore \,\,\lambda = - 7,\, - 6,\,......,\,8. \cr} $$