Page 64 - Standard Handbook Of Petroleum & Natural Gas Engineering

Analytic Geometry 53

Equations of a Circle (Center (h,k))

(x - h)2 + (y - k)2 = r2
Origin at center
x2 + y2 = rz
General equation
x2 + y2 + Dx + Ey + F = 0
where center = (-D/2, -E/2)
radius = d( D/2)' + (E/2)' - F

Tangent to circle at (x,,y,)
1 1
X,X + yIy + -D(x + x,)+ - E(y + y,) + F = 0
2 2
Parametric form, replacing x and y by
x = a cos u
and
y = a sin u
Equations of a Parabola (Figure 1-40)

A parabola is the set of points that are equidistant from a given fixed point
(the focus) and from a given fixed line (the directrix) in the plane. The key
feature of a parabola is that it is quadrilateral in one of its coordinates and
linear in the other.
(y - k)' = ~P(X - h)
Coordinates of the vertex V(h,k) and of the focus F(h + p,k)

Y

Figure 1-40. Equation of a parabola.