Question

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

• A. $25<k<29$
• B. $9<k<29$
• C. $9<k<25$
• D. $5<k<25$

Answer 1

The correct option is A $25<k<29$


The equation of the circle is $x^2+y^2-6x-10y+k=0$
Centre, $C\equiv (3,5)$
Radius, $r=\sqrt{3^2+5^2-k}=\sqrt{34-k}$
Since the circle does not touch or intersect the $x-$axis,
$\therefore r<y-$coordinate of centre $C$
or, $\sqrt{34-k}<5$
$\Rightarrow 34-k<25$
$\Rightarrow k>9\cdots \left(1\right)$

Also, given circle does not touch or intersect the $y-$axis.
$\therefore r<x-$coordinate of centre $C$
or, $\sqrt{34-k}<3$
$\Rightarrow 34-k<9$
$\Rightarrow k>25\cdots \left(2\right)$

Since point $(1,4)$ is inside the circle, then its distance from centre $C<r$,
or, $\sqrt{(3-1{)}^2+(5-4{)}^2}<\sqrt{34-k}$
$\Rightarrow 5<34-k$
$\Rightarrow k<29\cdots \left(3\right)$
Taking intersection of $\left(1\right),\left(2\right),\left(3\right)$,
Range of $k$ is $25<k<29.$