Question

The number of integral values of λ for which the equation x² + y² + λx + (1 − λ) 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 1

According to the question:
$$\begin{gathered} \sqrt{{\left(\frac{-\lambda }{2}\right)}^2+{\left(\frac{\lambda -1}{2}\right)}^2-5}\le 5 \\ \Rightarrow {\left(\frac{-\lambda }{2}\right)}^2+{\left(\frac{\lambda -1}{2}\right)}^2\le 30 \end{gathered}$$
$$\begin{gathered} {\lambda }^2+{\left(\lambda -1\right)}^2\le 120 \\ \Rightarrow 2{\lambda }^2-2\lambda -119\le 0 \\ \mathrm{Using}\mathrm{quadratic}\mathrm{formula}: \\ \Rightarrow \lambda =\frac{2\pm \sqrt{2^2-4\left(2\right)\left(-119\right)}}{2\left(2\right)} \\ \Rightarrow \lambda =\frac{2\pm \sqrt{956}}{4} \\ \Rightarrow \lambda =\frac{1\pm \sqrt{239}}{2} \\ \Rightarrow \lambda =-7.23,8.23 \\ \Rightarrow -7.23\le \lambda \le 8.23 \\ \Rightarrow \lambda =-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8\left(\mathrm{If}\lambda \in \mathrm{\mathbb{Z}}\right) \end{gathered}$$

Thus, the number of integral values of $\lambda$ is 16.