## Monday, April 28, 2008

### Conic Sections - Parabola

Conic section

Definition: the locus of a moving point P such that its distance from a fixed point called focus bears a constant ratio 'e' to its distance from a fixed line called directinx is a called a conic section.

The ratio e is called eccentricity of the conic.

e = Distance of a point (P) from focus/distance of the point (P) from directrix

this conic section is called

parabola is e = 1.
ellipse if e<1. and
hyperbola if e>1.

Parabola

Equation y² = 4ax

Some important results connected with y² = 4ax

1. Vertex A(0,0)
2. Focus S (a,0)
3. Foot of the directrix Z (-a,0)
4. Equation of directrix x = -a
5. Equation of latus rectum (L.R.) x = a
6. Equation of axis y = 0
7. Equation of tangent at vertex x = 0
8. Length of latus rectum = 4a
9. Extremities of L.R = (a,2a) and (a, -2a)
10. Focal distance of any point (x,y) is a+x

For parabola equation x² = 4ay

1. Vertex A(0,0)
2. Focus S (0,a)
3. Foot of the directrix Z
4. Equation of directrix y = -a
5. Equation of latus rectum (L.R.)
6. Equation of axis x = 0
7. Equation of tangent at vertex y = 0
8. Length of latus rectum = 4a
9. Extremities of L.R = (2a,a) and (2a, -a)
10. Focal distance of any point (x,y) is

Other forms: y² = -4ax
x² = -4ay

The general form of a parabola with axis parallel to x axis is of the form

ay² + by + c = x

It reduces to the form

(y-k) ² = 4A(x-h)
Points on a parabola y² = 4ax
If (x1, y1) are points on a parabola

y1² = 4ax1
or x1 = y1²/4a

Hence points are on a parabola can be expressed as (y1²/4a,y1)

Equation of a chord joining two points y1 and y2.

The points will be (y1²/4a,y1), (y2²/4a,y2)

y(y1+y2)-4ax-y1y2 = 0

If the chord is a focal chord, it passes through focus which is (a,0)

Hence the condition is y1y2 = -4a²

Equation of a tangent at the point (x1,y1)

yy1 = 2a(x+x1)

Equation of a tangent at the point (x1,y1) in terms of slope

M = 2a/y1

y = mx+a/m

Condition for the line y = mx+c to be a tangent to the parabola y² = 4ax

c = a/m is the condition

Hence the line y = mx+a/m is always a tangent to the parabola y² = 4ax and its point of contact is (a/m²,2a/m)

Normal to the parabola y² = 4ax

y-y1 = -y1/2a(x-x1)

Normal in terms of slope
m = -y1/2a

Hence equation is
y = mx-2am-am³

Points of intersection of two tangents to y² = 4ax
Tangents are
yy1 = 2a(x+x1)
yy2 = 2a(x+x2)

[y1y2/4a,(y1+y2)/2]

Perpendicular tangents to a parabola intersect on the directrix.

From any point (h,k) three normals can be drawn to a parabola