Conic Sections - Ellipse by ednexa.com

1. Ellipse
1. Equation of the tangent to the ellipse
i.
at P (x1, y1) is
ii.
at P (θ) is
x 2 y2

1
a 2 b2
xx1 yy1
 2 1
a2
b
x
y
cos   sin   1
a
b
iii. in terms of slope m is y = mx 
point of contact is P   a m , b  .


2

c
c 
at P (θ) is
x 2 y2
 1
a 2 b2
at P (x1,y1) is
ii.
and
2
2. Equation of the normal to the ellipse
i.
a 2m2  b2
a 2 x b2 y

 a 2  b2
x1
y1
ax
by

 a 2  b2 .
cos  sin 
3. Equation of the director circle of the ellipse is x2
+ y2 = a2 + b2.
4. If the tangent at P on the ellipse meets the
directrix in F, then PF subtends a right angle at
the corresponding focus.

2. 5. The tangent and normal at any point of the
ellipse bisect the external and internal angles
between the focal radii to that point.
6. The product of the lengths of perpendicular
segments from the foci an any tangent to the
ellipse x  y  1 is b2.
2
2
a2
b2
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