Q88 / 160 · AP EAMCET 2024 (20 May Shift 2) · Mathematics
The circle $x^2+y^2-8 x-12 y+\alpha=0$ lies in the first quadrant without touching the coordinate axes. If $(6,6)$ is an interior point to the circle, then
- A$4 \lt \alpha \lt 6$
- B$6 \lt \alpha \lt 16$
- C$16 \lt \alpha \lt 48$
- Correct answerD$36 \lt \alpha \lt 48$
Solution
$(6,6)$ is an interior point of $x^2+y^2-8 x-12 y+\alpha=0$
$\Rightarrow 36+36-48-72+\alpha \lt 0 \Rightarrow \alpha \lt 48$
Also, radius $ \lt x$ and $y$ coordinate of centre $\Rightarrow \sqrt{16+36-\alpha} \lt 4$ and 6 $\Rightarrow 52-\alpha \lt 16$ and $36 \Rightarrow \alpha \geqslant 36 \therefore 36 \lt \alpha \lt 48$
Also, radius $ \lt x$ and $y$ coordinate of centre $\Rightarrow \sqrt{16+36-\alpha} \lt 4$ and 6 $\Rightarrow 52-\alpha \lt 16$ and $36 \Rightarrow \alpha \geqslant 36 \therefore 36 \lt \alpha \lt 48$
Hence, the correct answer is option: 'D'.