Quadratics - the Australian Mathematical Sciences Institute

A guide for teachers – Years 11 and 12 • {33}
after a little algebra. Also,
ST 2 = (a + ap 2 ) 2 = a 2 (1 + p 2 ) 2
and so SP = ST . Hence ∆ST P is isosceles and so ∠ST P = ∠SPT .
Thus the reflection property tells us that any ray
parallel to the axis of the parabola will bounce off
the parabola and pass through the focus. Conversely,
any ray passing through the focus will reflect
off the parabola in a line parallel to the axis
of the parabola (so that light emanating from the
focus will reflect in a straight line parallel to the
axis).
S
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Conic sections
The parabola is one of the curves known as the conic sections, which are obtained when
a plane intersects with a double cone.
The non-degenerate conic sections are the parabola, ellipse, hyperbola and circle.
Parabola
y
y2 = 4ax
Ellipse
y
b
x 2
y 2
a + = 1
2 b 2
0
x
–a a
0
x
–b
Hyperbola
y
–a a
0
x 2
y 2
a – 2 b = 1 2
x
Circle
y
a x2 + y2 = a2
–a a
x
–a