Not sure I've ever heard the term "Conic Equation". Anyway...
First, you should come to grips with the fact that there may be no point of intersection.
After that, a little quick homework can narrow things down quickly. The first, centered at (3,-9) and reaching 4 or 8, depending which way you go, might suggest that it does run into the second, whcih is centered at (9,-2) and reaches 3 or 7, depending on direction.
Next, that ellipse doesn't get anywhere near Quadrant II, so we can just ignore that piece of the hyperbola.
Hyperbola's asymptotes are pretty clear at
y = 2x+3 and y = 15-2x
Again, the ellipse doesn't get all that close to y = 2x+3, so I'm guessing that a new problem is in order. Find the points of intersection between the ellipse and y = 15-2x. This WILL tell you where to start looking. One should be closer than the other, it is an asymptote.