Question

The number of integral values of $\lambda$ for which $x^2 + y^2 + \lambda x + (1 - \lambda )y + 5 = 0$ is the equation of a circle whose radius cannot exceed $5$, is

• A. 14
• B. 18
• C. 16
• D. None

Answer 1

Answer: (C)

16

Answer 2

Radius $\leq 5$ $\sqrt{\frac{\lambda^{2}}{4}+\frac{(1-\lambda)^{2}}{4}-5} \leq 5$ $\Rightarrow \lambda^{2}+(1-\lambda)^{2}=20 \leq 100$ $\Rightarrow 2 \lambda^{2}-2 \lambda-119 \leq 0$ $\therefore \frac{1-\sqrt{239}}{2} \leq \lambda \leq \frac{1+\sqrt{239}}{2}$ $\Rightarrow-7.2 \leq \lambda \leq 8.2$ $\therefore \lambda=-7,-6,-5, \ldots \ldots \ldots 7,8$ in all $16$ values