Question

If circle $x^2+y^2-6x-10y+c=0$ does not touch (or) intersect the coordinates axes and the point $(1,4)$ is inside the circle, then the range of $c$ is

• A. $R-(6,25)$
• B. $(6,25)$
• C. $(25,29)$
• D. $(6,29)$

Answer 1

The correct option is C $(25,29)$

Given circle $S:x^2+y^2-6x-10y+c=0$
$\because (1,4)$ lies inside of the given circle
$\therefore S_1<0\Rightarrow 1^2+4^2-6-10\left(4\right)+c<0\Rightarrow c<29\cdots \left(1\right)$
Circle will not meet the $x-$axis if $g^2-c<0$
$\Rightarrow c>9\cdots \left(2\right)$
Circle will not meet the $y-$axis if $f^2-c<0$
$\Rightarrow c>25\cdots \left(3\right)$
From $\left(1\right),\left(2\right),\left(3\right)$ we get
$c\in (25,29)$