Question

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

 y² + 4x + 4y − 3 = 0 

Answer 1

Given:
y² + 4y + 4x −3 = 0 

\[\Rightarrow \left( y + 2 \right)^2 - 4 + 4x - 3 = 0\]
\[ \Rightarrow \left( y + 2 \right)^2 = - 4\left( x - \frac{7}{4} \right)\] 

Let \[Y = y + 2\] 

\[X = x - \frac{7}{4}\] 

Then, we have: \[Y^2 = - 4X\] 

Comparing the given equation with \[Y^2 = - 4aX\] \[4a = 4 \Rightarrow a = 1\] 

∴ Vertex = (X = 0, Y = 0) = \[\left( x = \frac{7}{4}, y = - 2 \right)\] 

Focus = (X = −a, Y = 0) = \[\left( x - \frac{7}{4} = - 1, y + 2 = 0 \right) = \left( x = \frac{3}{4}, y = - 2 \right)\] 

Equation of the directrix:
X = a
i.e.\[x - \frac{7}{4} = 1 \Rightarrow x = \frac{11}{4}\] 

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

Length of the latus rectum = 4a = 4 units