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in Parabola by (65.5k points)

Write the characteristics of the following parabolas. 

(a) y2 = 16x 

(b) y2 = -8x 

(c) 3x2 = -8y 

(d) x2 = 8y

1 Answer

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(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)

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