Question

Find the coordinates of the vertex and focus, equation of the directrix and axis of the following parabolas. 

i) y² + 4x + 4y – 3 = 0

ii) x² – 2x + 4y – 3 = 0 

Answer 1

i)Given parábola is y² + 4x + 4y – 3 = 0 

⇒ y² + 4y = –4x + 3 

⇒ (y + 2)² - 4 = –4x + 3 

⇒ (y + 2)² = –4x + 7 

h=7/4, k=-2,a=1

Equation of the directrix: x – h – a = 0 

x-7/4-1=0⇒4x-11=0

Equation of the axis is: y – k = 0 ⇒ y + 2 = 0 

ii) Given parábola is x² – 2x + 4y – 3 = 0 

 ⇒ x² – 2x = –4y + 3 

⇒ (x – 1)² - 1 = –4y + 3 

⇒ (x – 1)² = –4y + 4 

⇒ (x – 1)² = –4[y – 1] 

 h = 1, k = 1, a = 1 

Vertex A(h, k) = (1, 1) 

Focus (h, k – a) = (1, 1 – 1) = (1, 0) 

Equation of the directrix: y – k – a = 0 

 y – 1 – 1 = 0 ⇒ y – 2 = 0 

Equation of the axis is, x – h = 0 ⇒ x – 1 = 0.