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The Circle — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSThe Circle1 Mark
Question
If the point $(\lambda,\ \lambda+1)$ lies inside the region bounded by the curve $\text{x}=\sqrt{25-\text{y}^2}$ and y-axis, then $\lambda$ belongs to the interval:
  1. $(-1,\ 3)$
  2. $(-4,\ 3)$
  3. $(-\infty,\ -4)\cup(3,\ \infty)$
  4. None of these
✓

Answer

  1. $(-1,\ 3)$

Solution:

The given equation of the curve is x2 + y2 = 25

Since $(\lambda,\ \lambda+1)$ lies inside the region bounded by the curve x2 + y2 = 25 and the y-axis, we have:

$\lambda^2+(\lambda+1)^2<25,$ provided $\lambda+1>0$

$\Rightarrow\lambda^2+\lambda^2+12\lambda<25,\ \lambda>-1$

$\Rightarrow2\lambda^2+2\lambda-24<0,\ \lambda>-1$

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

$\Rightarrow(\lambda-3)(\lambda+4)<0,\ \lambda>-1$

$\Rightarrow-4<\lambda<3,\ \lambda>-1$

$\Rightarrow\lambda\in(-1,\ 3)$

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