Question

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

y² − 4y + 4x = 0 

Answer 1

 Given:
y² − 4y + 4x = 0 

\[\Rightarrow \left( y - 2 \right)^2 - 4 + 4x = 0\]
\[ \Rightarrow \left( y - 2 \right)^2 = - 4\left( x - 1 \right)\] 

Let \[Y = y - 2\] 

\[X = x - 1\] 

Then, we have: 

\[Y^2 = - 4X\] 

Comparing the given equation with \[Y^2 = - 4aX\] 

\[4a = 4 \Rightarrow a = 1\]  

∴ Vertex = (X = 0, Y = 0) = \[\left( x = 1, y = 2 \right)\] 

Focus = (X = −a, Y = 0) = \[\left( x - 1 = - 1, y - 2 = 0 \right) = \left( x = 0, y = 2 \right)\] 

Equation of the directrix:
X = a 
i.e. \[x - 1 = 1 \Rightarrow x = 2\] 

Axis = Y = 0
i.e. \[y - 2 = 0 \Rightarrow y = 2\] 

Length of the latus rectum = 4a = 4 units