The set of values of k, for which the circle C:4x^2+4y^2-12x+8y+k=0 lies inside the fourth…

Question

The set of values of $k$, for which the circle $C:4x^2+4y^2-12x+8y+k=0$ lies inside the fourth quadrant and the point $\left(1,-\dfrac13\right)$ lies on or inside the circle $C$, is:

Accepted Answer

$C:x^2+y^2-3x+2y+\dfrac k4=0$, centre $\left(\dfrac32,-1\right)$, $r^2=\dfrac{13}4-\dfrac k4$.

Point on/inside: $4+\dfrac49-12-\dfrac83+k\le0\Rightarrow k\le\dfrac{92}9$.

Circle entirely in the fourth quadrant needs $r<1$ (distance to the nearer axis): $\dfrac{13}4-\dfrac k4<1\Rightarrow k>9$.

$$k\in\left(9,\frac{92}9\right].$$

Solutions · 1