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If the Point (λ, λ + 1) Lies Inside the Region Bounded by the Curve \[X = \Sqrt{25 - Y^2}\] And Y-axis, Then λ Belongs to the Interva - Mathematics

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प्रश्न

If the point (λ, λ + 1) lies inside the region bounded by the curve \[x = \sqrt{25 - y^2}\] and y-axis, then λ belongs to the interval

विकल्प

  • (−1, 3)

  • (−4, 3)

  • (−∞, −4) ∪ (3, ∞)

  • none of these

MCQ
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उत्तर

(−1, 3)

The given equation of the curve is \[x^2 + y^2 = 25\].

Since (λ, λ + 1) lies inside the region bounded by the curve

\[x^2 + y^2 = 25\] and the y-axis, we have:

\[\lambda^2 + \left( \lambda + 1 \right)^2 < 25\],
\[\text { provided } \lambda + 1 > 0\]

\[\Rightarrow \lambda^2 + \lambda^2 + 1 + 2\lambda < 25, \lambda > - 1\]

\[ \Rightarrow 2 \lambda^2 + 2\lambda - 24 < 0, \lambda > - 1\]

\[ \Rightarrow \lambda^2 + \lambda - 12 < 0, \lambda > - 1\]

\[ \Rightarrow \left( \lambda - 3 \right)\left( \lambda + 4 \right) < 0, \lambda > - 1\]

\[ \Rightarrow - 4 < \lambda < 3, \lambda > - 1\]

\[ \Rightarrow \lambda \in \left( - 1, 3 \right)\]

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Advanced Concept of Circle - Standard Equation of a Circle
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 24: The circle - Exercise 24.6 [पृष्ठ ३९]

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आरडी शर्मा Mathematics [English] Class 11
अध्याय 24 The circle
Exercise 24.6 | Q 10 | पृष्ठ ३९

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