Page 57 - Aeronautical Engineer Data Book

44 Aeronautical Engineer’s Data Book
Circle
2
2
General equation x – y + 2gx + 2fy + c = 0
The centre has co-ordinates (–g, –f)
The radius is r = �g + f � 2 – c
�
2
The equation of the tangent at (x , y )
1
1
to the circle is:
+ yy + g(x + x ) + f(y + y ) + c = 0
xx 1 1 1 1
The length of the tangent from to the circle is:
2
2
2
t = x + y + 2gx + 2fy + c
1
1
1
1
Parabola (see Figure 2.10)
SP
Eccentricity = e = 33 = 1
PD
With focus S(a, 0) the equation of a parabola
2
is y = 4ax.
The parametric form of the equation is x =
2
at , y = 2at.
, y ) is yy
The equation of the tangent at (x 1 1 1
= 2a(x + x ).
1
Ellipse (see Figure 2.11)
SP
Eccentricity e = 33 < 1
PD
x 2 y 2
The equation of an ellipse is 33 + 33 = 1
a 2 b 2
2
2
2
where b = a (1 – e ).
The equation of the tangent at (x , y ) is
1
1
xx 1 yy 1
33 + 33 = 1.
a 2 b 2
The parametric form of the equation of an
ellipse is x = a cos , y = b sin , where is
the eccentric angle.
Hyperbola (see Figure 2.12)
SP
Eccentricity e = 33 > 1
PD
x 2 y 2
The equation of a hyperbola is 33 – 33 = 1
a 2 b 2
2
2
2
where b = a (e – 1).