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Conic Sections — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsConic Sections1 Mark
MCQ
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:
  • ✓
    $(-1,\ 3)$
  • B
    $(-4,\ 3)$
  • C
    $(-\infty,\ -4)\cup(3,\ \infty)$
  • D
    None of these
✓

Answer

Correct option: A.
$(-1,\ 3)$
The given equation of the curve is $x^2+ y^2= 25$
Since $(\lambda,\ \lambda+1)$ lies inside the region bounded by the curve $x^2+ y^2= 25$ 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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