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)

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