Find the radical axis of the circles $x^2+y^2-6x+8y-12=0,\text{ and }{x}^2+y^2+6x-8y+12=0$. Prove that the radical axis is perpendicular to the line joining the centres of the two circles.

If centre of one circle lies on or inside another, prove that the circles cannot be orthogonal.

If a circle $S$ intersects circles $S_1\text{ and }{S}_2$ orthogonally, prove that the centre of $S$ on the radical axis of $S_1\text{ and }{S}_2$.

$R$ is the radical centre of circles $S_1,S_2\text{ and }{S}_3$. Prove that if $R$ is on / inside / outside one of the circles then it is similarly situated with respect to the other two.

Determine a circle which cuts orthogonally each of the circles: $S_1:x^2+y^2-4x-6y+12=0,S_2:x^2+y^2+4x+6y+12=0,S_3:x^2+y^2-4x+6y+12=0$.

Prove that no pair of concentric circles can have a radical axis.

The equation $y=x^2+2x+3$ represents a parabola with its axis parallel to $y-$axis.

The latus rectum of the parabola $y^2=-8x\text{ is }2$.

The eccentricity of the parabola $(y-1{)}^2=4(x+3)\text{ is }\frac{1}{3}$.