Find the directrices of

Question

1. 5x^2+6y^2=30

 

2. y^2-6x=12

 

3. 2x^2-y^2=4

Answer 1

vertical ellipses have directrix y=+/-b/e

horizontal ellipses have directrix x =+/-a/e where e=eccentricity

x^2/a^2 +y^2/b^2 = 1 is a standard form ellipse equation for a and b are semi axes

1) 5x^2 +6y^2 = 30 is equivalent to

x^2/6 + y^2/5 = 1

where a = sqr6 and b=sqr5 a>b, so the ellipse is horizontal with directrix x=+/-(sqr6)/e, e=c/a where c=distance from center to a focus e=sqr6/6

directrices are x=6 and x=-6

solve 3) in a similar way

x=+/-(sqr6)/3 are the two directrices

2) has the more well known directrix, a line equidistant from the vertex to the focus

it's a horizontal parabola with a vertical directrix

y^2-6x = 12

one directrix

x = -7/2 = -3.5