MCQ
The circle $x^2+ y^2+ 2gx + 2fy + c = 0$ does not intersect $x-$axis, if:
- ✓$g^2 < c$
- B$g^2 > c$
- C$g^2 > 2c$
- DNone of these
✓
Answer
Correct option: A.
$g^2 < c$
Given:
$x^2+ y^2+ 2gx + 2fy + c = 0 ......... (1)$
The given circle intersects the $x-$axis.
The equation of circle becomes $x^2+ 2gx + c = 0 ......... (2)$
Solving equation $(2):$
$\therefore$ Discriminant, $\text{D}=\sqrt{4\text{g}^2-4\text{c}}\geq0$
$\Rightarrow4\text{g}^2-4\text{c}\geq0$
$\Rightarrow\text{g}^2-\text{c}\geq0$
$\Rightarrow\text{g}^2\geq\text{c}$
Hence, if $g^2 < c$, then the given circle will not intersect the $x-$axis.
$x^2+ y^2+ 2gx + 2fy + c = 0 ......... (1)$
The given circle intersects the $x-$axis.
The equation of circle becomes $x^2+ 2gx + c = 0 ......... (2)$
Solving equation $(2):$
$\therefore$ Discriminant, $\text{D}=\sqrt{4\text{g}^2-4\text{c}}\geq0$
$\Rightarrow4\text{g}^2-4\text{c}\geq0$
$\Rightarrow\text{g}^2-\text{c}\geq0$
$\Rightarrow\text{g}^2\geq\text{c}$
Hence, if $g^2 < c$, then the given circle will not intersect the $x-$axis.
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