Finding number of values of slope of line L

Question

The following question is taken from JEE practice set.

In a plane rectangular coordinate system, there are three points $A(0,\frac43), B(-1,0)$ and $C(1,0)$. The distance from point P to line BC is geometric mean of the distances from this point to lines AB and AC. If line L passes through the incenter D of triangle ABC, and has exactly 3 common points with the locus of point P, find the number of values of slope of line L.

I found the equations of lines AB, BC, CA. Then found the distances of P from these lines. Using the given condition, I got the locus of P as

$2x^2+2y^2+3y-2=0$ or $8x^2-17y^2+12y-8=0$

The locus is either a circle or a hyperbola.

If a line is to pass through the incenter and meet the locus, can it have 3 points in common?

Or is the question implying that we need to consider both the loci simultaneously?

Edit: The answer given is $7$.

Answer 1

The locus of $P$ is formed by the points of both curves, circle and hyperbola. If a line has three points in common with the locus, then either it is tangent to one of the curves, or it passes through a point where the curves intersect ($B$ or $C$).