Conics Definition

1 Conic Equation

A conic section is the locus of points where the proportion of the distance from any of the points to a fixed point, called the focus and the perpendicular distance from the same point to a straight line, called the directrix, is a fixed number e, called the excentricity.

Suppose we set the focus at the point ( a , 0 ), where the section passes through the point ( 0 , 0 ), so the directrix is the straight line parallel to the y-axis at ...a ⁄ e ...from the axis.

In the diagram (right) we have ......SP ...= ...e PM . ......So

( a – x )² + y² ...= ...SP² ...= ...e² PM²
..................= ...e² ( a ⁄ e ...+ ...x )² ...= ...( a + e x )²

y² ...= ...a² + e²x² + 2 a e x – a² – x² + 2 a x

......y2 ...= ...( e2 – 1 ) x2 + 2 a ( e + 1 ) x

2 Parametric Equation

We can also describe the curve in terms of the angle θ as shown on the diagram

( a – x ) ...= ...PS cos θ ...= ...e PM cos θ ...= ...( a + ex ) cos θ

x ...= ... a ( 1 – cos θ ) 1 + e cos θ

y ...= ...PS sin θ ...= ...( a + ex ) sin θ

............= ...

	( a + ae cos θ ) + ae ( 1 – cos θ )
	1 + e cos θ

sin θ

y ...= ... a ( 1 + e ) sin θ 1 + e cos θ

3 Conic Types

So far the only restrictions we have placed on the parameters, albeit implicitly, is that a is positive, and e is non-negative.

y² ...= ...( e² – 1 ) x² + 2 a ( e + 1 ) x , ......a > 0 , ...e ≥ 0 ,

consider the expression

z ...= ...( e² – 1 ) x² + 2 a ( e + 1 ) x

which we can write

z ...= ...p x² + q x ...= ...x ( p x + q )

where ......p = ( e² – 1 ), ...q = 2 a ( e + 1 ), ...so ...-1 ≤ p < ∞ ,...and ...0 ≤ q < ∞

Now y has real roots only if z is non-negative, which is the case if:

x ≥ 0 ...and ...( p x + q ) ≥ 0 ⇔ { x ≥ – q ⁄ p , p positive | x ≤ – q ⁄ p , p negative } ......(1)

x ≤ 0 ...and ...( p x + q ) ≤ 0 ⇔ { x ≤ – q ⁄ p , p positive | x ≥ – q ⁄ p , p negative } ......(2)

[Note that q is always positive, and p is positive if e > 1.]

So, if p > 0, y has real roots if

x > 0 ...(1) ......or ......x < – q ⁄ p ...(2)

If p < 0, y has real roots if

0 < x < – q ⁄ p ...(1) ......or ......– q ⁄ p < x < 0 ...(2) – contradiction as – q ⁄ p > 0

If p = 0 then ( p x + q ) = q which is always positive, so y has positive roots only if x >0.

3.1 Hyperbola

p is greater than zero when e is greater than 1. In this case, as shown above, y has real roots in the region

......x ...< ...-q ⁄ p ...= ...– 2 a ( e + 1 ) ⁄ ( e² – 1 ) ...= ...– 2 a ⁄ ( e – 1 )

and the region

......x ...> ...0

When e is greater than 1 the curve is known as a hyperbola

 

3.2 Parabola

3.3 Elipse

p is less than zero when e is less than 1. In this case y has real roots in the region

......0 ...< ...x ...< ...-q ⁄ p ...= ...2 a ⁄ ( 1 – e )

When e is less than 1 the curve is known as an elipse

 

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