Question

Write the characteristics of the following parabolas.

(a) y² = 8x 

(b) y² + 4x = 0 

(c) 3x² + 4y = 0 

(d) x² + 4x = 0

Answer 1

(a) y² = 8x compare with y² = 4ax. 

The curve turns right side

4a = 8 ⇒ a = 2 

Vertex, V = (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

Length of LR 4a = 8 

Ends of LR (2, 4) (2,-4) 

(b) Compare y² = -4x with y² = -4ax. 

The curve turns

Left hand side 4a = 4 ⇒ a = 1 

Vertex, V = (0,0) 

Axis x-axis (equation is y = 0) 

Tangent y-axis (equation is x = 0) 

Directrix, x = 1 or x = -1 = 0 

Equation of LR x = -1 or x + 1 = 0 

Ends of LR (-1,2) (-1, -2) 

Length of LR = 4a = 4 

(c) 3x² + 4y = 0

x² = \(\frac{-4}{3}\)y compare with x² = – 4ay. 

The curve turns downwards and 4a = \(\frac{4}{3}\)⇒ a = \(\frac{1}{3}\) 

Vertex, V = (0, 0) 

Axis y-axis (equation is x = 0) 

Tangent x-axis (equation is y = 0)

Focus, s=(0,- \(\frac{1}{3}\)) 

Directrix y = \(\frac{1}{3}\) or 3y – 1= 0 

Equation of LR y = \(\frac{-1}{3}\) or 3y +1 = 0 

Length of LR = 4a = \(\frac{4}{3}\) 

Ends of LR (- \(\frac{2}{3}\), - \(\frac{1}{3}\)) (\(\frac{2}{3}\), - \(\frac{1}{3}\))

(d) x² + 16y = 0 compare with x² = -4ay

x² = -16y. 

The curve turns downward 

4a = 16 ⇒ a = 4 

Vertex, V = (0,0) 

Axis y-axis (Equation is x = 0) 

Tangent x-axis (Equation is y = 0) 

Focus, S = (0, -4) 

Directrix y = 4 or y – 4 = 0 

Equation of LR y = -4 or y + 4 = 0 

Length of LR = 4a = 16 

Ends of LR (-8,-4) (8,-4).