Conics

Question 1

Parabola focus directrix definition

Answer 1

A parabola is the locus of points from a fixed point S (focus) and a fixed straight line (directrix)

for y^2 = 4ax
focus (a,0) directrix - x=-a vertex at (0,0)

Question 2

eq of standard hyperbola cartesian

Answer 2

(x/a)²+(y/b)² = 1

as x and y tend to inf, (x/a)² approx (y/b)²
so y = +-bx/a

Question 3

Parametric eq of hyperbola

Answer 3

x = +-acosh(t) y = bsinh(t)
or
x = asec(t) y = btan(t)
t isnt pi/2, between pi n -pi

Question 4

Eccentricity of a curve e

Answer 4

Ratio of distance from a fixed point P (focus) and a fixed straight line (directrix) is constant
0<e<1 - Elipse
e=1 - parabola
1<e - hyperbola
e=0 - circle
e=infinity - straight line

Question 5

remember for eccentricity, foci n directrices

Answer 5

foci are on the major axis, for standard eq, if a>b foci at (+-ae,0) directrices x=+-a/e

b²=a²(1-e²) swap if b>a

Question 6

Actual useful results

Answer 6

PS + PS’ = 2a for a>b an elipse