Single Choice

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$$16$$
Correct Answer
DNone

Solution

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

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