Question

Find the vertex, focus, axis, directrix and latus-rectum of the following parabola

y² = 8x + 8y 

Answer 1

Given:
y² = 8x + 8y 

\[\Rightarrow \left( y - 4 \right)^2 = 8\left( x + 2 \right)\] 

Putting \[Y = y - 4\] 

\[X = x + 2\] 

\[Y^2 = 8X\] 
On comparing the given equation with \[Y^2 = 4aX\]

\[4a = 8 \Rightarrow a = 2\] 

∴ Vertex = (X = 0, Y = 0) = \[\left( x = - 2, y = 4 \right)\] 

Focus = (X = a, Y = 0) = \[\left( x + 2 = 2, y - 4 = 0 \right) = \left( x = 0, y = 4 \right)\]

Equation of the directrix:
X = −a 
i.e. \[x + 2 = - 2 \Rightarrow x + 4 = 0\] 

Axis = Y = 0
i.e. \[y -  4 = 0 \Rightarrow y = 4\]

Length of the latus rectum = 4a = 8