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  • The number of integral values of for which <img alt="x^2+y^2+\lambda x+(1-\lambda) y+5=0" src="https://entrance

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

Option: 1

14


Option: 2

18


Option: 3

16


Option: 4

None of these

 


Answers (1)

best_answer

Centre of circle =\left(-\frac{\lambda}{2},-\frac{(1-\lambda)}{2}\right)     ; Radius of circle =\sqrt{\left(\frac{\lambda}{2}\right)^2+\left(\frac{1-\lambda}{2}\right)^2-5} \leq 5

\Rightarrow \quad 2 \lambda^2-2 \lambda-119 \leq 0                   \therefore \quad \frac{1-\sqrt{239}}{2} \leq \lambda \leq \frac{1+\sqrt{239}}{2}

\Rightarrow-7.2 \leq \lambda \leq 8.2 \quad \text { (Nearly). }                   \therefore \lambda=-7,-6, \quad \ldots \ldots \ldots \ldots, 7,8

Hence number of integral values of \lambda is 16.

 

 

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vinayak

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