Conic sections

An ellipse

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var('x y')
gr = implicit_plot((x/5)^2 + (y/3)^2 == 1, (x,-5,5), (y,-5,5)) #Change the equation. Have fun!
show(gr, axes=True, aspect_ratio=1)

A parabola

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var('x y')
gr = implicit_plot((x/5)^2 - (y/3) == 0, (x,-5,5), (y,-5,5))
show(gr, axes=True, aspect_ratio=1)

A hyperbola

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var('x y')
gr = implicit_plot((x/5)^2 - (y/3)^2 == 1, (x,-10,10), (y,-10,10))
show(gr, axes=True, aspect_ratio=1)

Family of ellipses

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var('x y')
gr = Graphics()
for i in [1..5]:
    gr += implicit_plot(x^2 + i*y^2 == 1, (x,-1,1), (y,-1,1), color = hue((i-1)/10))
show(gr, axes=True, aspect_ratio=1)

Family of parabolas

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var('x y')
gr = Graphics()
for i in [1..5]:
    gr += implicit_plot(i*y - x^2 == 0, (x,-3,3), (y,-3,3), color = hue((i-1)/10))
show(gr, axes=True, aspect_ratio=1)

Family of hyperbolas

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var('x y')
gr = Graphics()
for i in range(2,-3,-1):
    gr += implicit_plot(x^2 - y^2 == i, (x,-5,5), (y,-5,5), color = hue((2-i)/10))
show(gr, axes=True, aspect_ratio=1)

Note: The equation

x^{2} - y^{2} = 0

is that of a "degenerate hyperbola" consisting of the two lines

x + y = 0

and

x - y = 0