again, just replace x with rcos(theta) and y with rsin(theta) and i'm tired of typing theta, i'll use t from now on
x^2 + y^2 - 8y = 0
=> (rcos(t))^2 + (rsin(t))^2 - 8(rsin(t)) = 0
=> r^2*cos^2(t) + r^2*sin^2(t) - 8rsin(t) = 0
=> r^2*(cos^2(t) + sin^2(t)) - 8rsin(t) = 0
=> r^2 - 8rsin(t) = 0
and i'd leave it there
i find questions like these so annoying i don't know why. whenever i see "find the directrix and focus of a parabola", i cringe. anyway...
this looks fine
for the foci i got (0, +/- sqrt(84)/5) ........check my calculations. for (x^2)/(b^2) + (y^2)/(a^2) = 1 where a >= b > 0 the foci is (0, +/- c) where c^2 = a^2 - b^2
question 2
25x2+4y2-16=0
25x2/4^2+y2/2^2=1
a2+b1=c2 16+4=c2 2sqrt5=c therefore foci (+-2sqrt5,0) verticies(0,+-2)
for asymptote:
question 3
x2/25-y2/4=1
y2=4+(4x2/25)
y=+-(2sqrt(x2+25))/5
y = +/- (b/a)x = +/- (2/5)x