Ellipse

1. Equation of the tangent to the ellipse 2 2

2 2

x y1

a b

i. at P (x1, y1) is 1 1

2 2

xx yy1

a b

ii. at P (θ) is x ycos sin 1

a b

iii. in terms of slope m is y = mx 2 2 2a m b and

point of contact is P2 2a m b

, .c c

2. Equation of the normal to the ellipse 2 2

2 2

x y1

a b

i. at P (x1,y1) is 2 2

2 2

1 1

a x b ya b

x y

ii. at P (θ) is 2 2ax bya b .

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.

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

ellipse2 2

2 2

x y1

a b is b2.

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