Conic Sections - Ellipse

Ellipse
Equation x²/a² + y²/b² = 1

Some important results connected with x²/a² + y²/b² = 1

1. Vertex A(-a,0), A’(a,0)
2. Focus S
3. Foot of the directrix Z
4. Equation of directrix
5. Equation of latus rectum (L.R.)
6. Equation of axis
7. Equation of tangent at vertex
8. Length of latus rectum = 2b²/a
9. Extremities of L.R = (-ae,b²/a) and (-ae,-b²/a)
10. Focal distance of any point

the sum of th focal distances of a point on the ellipse is constant and is equal to the length of the major axis.


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

xx1/a² + yy1/b² = 1

Normal

x-x1/(x1/a²) = y-y1/(y1/b²)

Conormal points are those points, the normals at which pass through the same point. eg-P,Q,R are conormal points if normals at P,Q,R pass through the same point S.

Conidition for tangent

If y = mx+c is to be a tangent

C = ±SQRT(a²m²+b²)

Hence equation of a tangent

Y = mx±SQRT(a²m²+b²)

The points of contact

(-a²m/ SQRT(a²m²+b²), b²m/ SQRT(a²m²+b²)) or

(-a²m/c, b²/c)

Director circle

The locus of the point of intersection of two perpendicular tangents to an ellipse is called director circle.

The equation of director circle is

x² + y² = a²+b²

chord of contact points of the tangents drawn from the point (x1,y1)

xx1/a² + yy1/b² = 1

Polar of a point (x1,y1) w.r.t. ellipse is

xx1/a² + yy1/b² = 1

Polar is the locus of point, that is an intersection of tangents drawn from the extreme points of the chords which are drawn from a given point (x1,y1). The given point is called the pole of the polar.

Chord with a given middle point (h,k)

hx/a² + ky/b² = h²/a²+k²/b²

Diameter: Any line passing through the centre (0,0) of an ellipse is called a diameter hence its equation is of the form
y = mx.