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

विकल्प

• 14

• 18

• 16

• none of these

Answer 1

According to the question:

$$\sqrt{{\left(\frac{-\lambda }{2}\right)}^2+{\left(\frac{\lambda -1}{2}\right)}^2-5}\le 5$$

$$\Rightarrow {\left(\frac{-\lambda }{2}\right)}^2+{\left(\frac{\lambda -1}{2}\right)}^2\le 30$$

$${\lambda }^2+{(\lambda -1)}^2\le 120$$

$$\Rightarrow 2{\lambda }^2-2\lambda -119\le 0$$

Using quadratic formula: 

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

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

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

$$\Rightarrow \lambda =-7.23,8.23$$

$$\Rightarrow -7.23\le \lambda \le 8.23$$

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

Thus, the number of integral values of

$$\lambda$$ is 16.