What conic section is represented by the equation show below? 9x+4y^2+18x=16

Conic sections are cross-sectional areas of a cone (the 2-d figure you get when you slice a cone). You can get circle, an ellipse, a parabola, or a hyperbola. 

In this case it is a hyperbola because, when graphed, it is not a function (so not a parabola) and does not form a circle or ellipse/oval shape.

Please refer to the attached diagram. The given circle is the blue circle centered at A and the given exterior point is C.

Draw line segment ACUsing the same radius for each, which radius is longer than half the length of AC, draw the orange arcs centered at A and C so that they intersect at points D and E.Draw the yellow line DE. Designate the point where it intersects AC as point F. This is the midpoint of AC.Using F as the center, draw green circle F with radius FA (or FC). Designate the points where circle F intersects circle A as points G and H.Your tangent lines are CG and CH (shown in lavender).