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BITSAT 2016
Exam
BITSAT
Year
2016
Subject
Mathematics
Chapter
Circle

BITSAT 2016 · Mathematics · Circle

The number of integral values of λ for which ^ 2 + ^ 2 +λ x+(1-λ)...

The number of integral values of λ\lambda for which \mathrmx2+\mathrmy2+λx+(1−λ)y+5=0\mathrmx^{2}+\mathrmy^{2}+\lambda \mathrm{x}+(1-\lambda) \mathrm{y}+5=0 is the equation of a ircle whose radius cannot exceed 5,5, is

Options

  1. A14
  2. B18
  3. C16
  4. DNone

Correct answer

C. 16

Step-by-step solution

Radius ≤5\leq 5λ24+(1−λ)24−5≤5⇒λ2+(1−λ)2−20≤100⇒2λ2−2λ−119≤0∴1−2392≤λ≤1+2392⇒−7.2≤λ≤8.2( approx. )∴λ=−7,−6,−5,…….7,8, in all 16\begin{array}{l} \sqrt{\frac{\lambda^{2}}{4}+\frac{(1-\lambda)^{2}}{4}-5} \leq5 \Rightarrow \lambda^{2}+(1-\lambda)^{2}-20 \leq 100 \\ \Rightarrow 2 \lambda^{2}-2 \lambda-119 \leq 0 \\ \therefore \frac{1-\sqrt{239}}{2} \leq \lambda \leq \frac{1+\sqrt{239}}{2} \Rightarrow-7.2 \leq \lambda \leq 8.2 \\ \quad(\text{ approx. })\\ \therefore \lambda=-7,-6,-5, \ldots \ldots .7,8, \text{ in all }16 \end{array}$$ values
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