(a) Compare y2 = 16x with y2 = 4ax
We get 4a = 16 ⇒ a = 4
The curve turns right side with vertex (0,0)
Focus S(a,0) = (4,0)
Directrix x = -a or x = -4 or x + 4 = 0
Axis x-axis (equation is y = 0)
Tangent y-axis (equation is x = 0)
Equation of LR is x = a ⇒ x = 4 or x – 4 = 0
Length of LR 4a = 16
Ends of LR (a, 2a) (a, -2a) = (4,8) and (4,-8)
(b) y2 = -8x compare with y2 = -4ax
The curve turns left hand side and 4a = 8 ⇒ a = 2
Vertex = (0,0)
Axis x-axis (equation is y = 0)
Tangent y-axis (equation is x = 0)
Focus S(-2,0)
Directrix, x = 2 or x – 2 = 0
Equation of LR x = -2 or x + 2 = 0
Length of LR = 4a = 8
Ends of LR (-a, 2a) (-a, -2a) = (-2, 4) (-2,-4)
(C) 3x2 = -8y ⇒ x2 = \(\frac{-8}{3}\)y compare this with
x2 = -4ay ⇒ 4a = \(\frac{8}{3}\)⇒ a = \(\frac{2}{3}\)
The curve turns downwards
Vertex V = (0,0)
Axis y-axis (equation is x = 0)
Focus S = (0, - \(\frac{2}{3}\))
Tangent x-axis (equation is y = 0)
Directrix y = \(\frac{2}{3}\) = or 3y – 2 = 0
Equation of LR y = - \(\frac{2}{3}\) – or 3y + 2 = 0
Length of LR = 42 = \(\frac{8}{3}\); Ends of LR (\(\frac{4}{3}\), - \(\frac{2}{3}\)) (- \(\frac{4}{3}\), - \(\frac{2}{3}\))
(d) x2 = 8y compare this with x2 = 4ay the curve turns upwards
4a = 8 ⇒ a = 2
Vertex, V = (0,0)
Axis y-axis (equation is x = 0)
Tangent x-axis (equation is y = 0)
Directrix y = -2 or y + 2 = 0
Equation of LR y = 2 or y – 2 = 0
Length of LR 4a = 8
Ends of LR (4, 2) (-4, 2)