The equation $(x - a)^2 + (y - b)^2 = k(lx + my + n)^2$ represents
Step-by-Step Solution
Key Concept: The ratio of distance from a point to a fixed point (focus) versus distance to a fixed line (directrix) determines conic type through the constant $k(l^2 + m^2)$.
The general conic equation $(x-a)^2 + (y-b)^2 = k(lx + my + n)^2$ represents different curves depending on the value of $k$. The eccentricity is determined by $e = \sqrt{k}\sqrt{l^2 + m^2}$ normalized by $\sqrt{l^2 + m^2}$. If $k(l^2 + m^2) = 1$, the curve is a parabola; if $k(l^2 + m^2) 1$, it's a hyperbola; and if $k = 0$, it's a point circle.
Correct Answer: 2,3,4