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

The correct option is C $16$

Given that $Radius\le 5$
$\sqrt{\frac{{\lambda }^2}{4}+\frac{(1-\lambda {)}^2}{4}-5}\le 5\Rightarrow {\lambda }^2+(1-\lambda {)}^2-20\le 100$
$\Rightarrow 2{\lambda }^2-2\lambda -119\le 0$
$\Rightarrow \frac{1-\sqrt{239}}{2}\le \lambda \le \frac{1-\sqrt{239}2}{\Rightarrow }-7.2\le \lambda \le 8.2\left(\text{approx.}\right)$
Therefore, $\lambda =-7,-6,-5,....,7,8$, in all $16$ values.