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

Options

• 14

• 18

• 16

• none of these

Answer 1

According to the question:

\[\sqrt{\left( \frac{- \lambda}{2} \right)^2 + \left( \frac{\lambda - 1}{2} \right)^2 - 5} \leq 5\]

\[ \Rightarrow \left( \frac{- \lambda}{2} \right)^2 + \left( \frac{\lambda - 1}{2} \right)^2 \leq 30\]

\[\lambda^2 + \left( \lambda - 1 \right)^2 \leq 120\]

\[ \Rightarrow 2 \lambda^2 - 2\lambda - 119 \leq 0\]

Using quadratic formula: 

\[ \Rightarrow \lambda = \frac{2 \pm \sqrt{2^2 - 4\left( 2 \right)\left( - 119 \right)}}{2\left( 2 \right)}\]

\[ \Rightarrow \lambda = \frac{2 \pm \sqrt{956}}{4}\]

\[ \Rightarrow \lambda = \frac{1 \pm \sqrt{239}}{2}\]

\[ \Rightarrow \lambda = - 7 . 23, 8 . 23\]

\[ \Rightarrow - 7 . 23 \leq \lambda \leq 8 . 23\]

\[ \Rightarrow \lambda = - 7, - 6, - 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5, 6, 7, 8 \left( If \lambda \in \mathbb{Z} \right)\]

Thus, the number of integral values of

\[\lambda\] is 16.