5. Conics & Parameters
3) Hyperbola y
P x, y
x2 y2 a2 a y
x x
x
sec
a
x a sec
6. Conics & Parameters
3) Hyperbola y
P x, y
x2 y2 a2 a y
x x
a 2 sec 2 y 2
x when x a sec , 2 1
sec
2
a b
a
x a sec
7. Conics & Parameters
3) Hyperbola y
P x, y
x2 y2 a2 a y
x x
a 2 sec 2 y 2
x when x a sec , 2 1
sec
2
a b
a y2
sec 2 1
x a sec b2
y 2 b 2 tan 2
y b tan
8. Conics & Parameters
3) Hyperbola y
P x, y
x2 y2 a2 a y
x x
a 2 sec 2 y 2
x when x a sec , 2 1
sec
2
a b
a y2
sec 2 1
x a sec y b tan b2
y 2 b 2 tan 2
y b tan
10. x2 y2
For hyperbola 2 2 1
a b
tangent at x1 , y1
x1 x y1 y
2
2 1
a b
11. x2 y2
For hyperbola 2 2 1
a b
tangent at x1 , y1 normal at x1 , y1
a 2 x b2 y
a 2 b2 a 2e2
x1 x y1 y
2
2 1
a b x1 y1
12. x2 y2
For hyperbola 2 2 1
a b
tangent at x1 , y1 normal at x1 , y1
a 2 x b2 y
a 2 b2 a 2e2
x1 x y1 y
2
2 1
a b x1 y1
tangent at a sec , b tan
x sec y tan
1
a b
13. x2 y2
For hyperbola 2 2 1
a b
tangent at x1 , y1 normal at x1 , y1
a 2 x b2 y
a 2 b2 a 2e2
x1 x y1 y
2
2 1
a b x1 y1
tangent at a sec , b tan normal at a sec , b tan
x sec y tan
a 2 b2 a 2e2
ax by
1
a b sec tan
14. x2 y2
For hyperbola 2 2 1
a b
tangent at x1 , y1 normal at x1 , y1
a 2 x b2 y
a 2 b2 a 2e2
x1 x y1 y
2
2 1
a b x1 y1
tangent at a sec , b tan normal at a sec , b tan
x sec y tan
a 2 b2 a 2e2
ax by
1
a b sec tan
y sec f x y tan f x
dy dy
f x sec f x tan f x f x sec 2 f x
dx dx
15. x2 y2
For hyperbola 2 2 1
a b
tangent at x1 , y1 normal at x1 , y1
a 2 x b2 y
a 2 b2 a 2e2
x1 x y1 y
2
2 1
a b x1 y1
tangent at a sec , b tan normal at a sec , b tan
x sec y tan
a 2 b2 a 2e2
ax by
1
a b sec tan
y sec f x y tan f x
dy dy
f x sec f x tan f x f x sec 2 f x
dx dx
Exercise 6C; 5, 7, 9, 16, 17