Question

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

 y² = 5x − 4y − 9 

Answer 1

Given: 

 y ² = 5x − 4y − 9 

\[\Rightarrow y^2 + 4y = 5x - 9\]
\[ \Rightarrow \left( y + 2 \right)^2 = 5x - 5 = 5\left( x - 1 \right)\]

Putting \[Y = y + 2\] 

\[X = x - 1\]

\[Y^2 = 5X\]

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

\[4a = 5 \Rightarrow a = \frac{5}{4}\]

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

Focus = (X = a, Y = 0) =\[\left( x - 1 = \frac{5}{4}, y + 2 =0 \right) = \left( x = \frac{9}{4}, y = - 2 \right)\] 

Equation of the directrix:
X = −a
i.e.\[x - 1 = \frac{- 5}{4} \Rightarrow x = \frac{- 1}{4}\] 

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

Length of the latus rectum = 4a = 5 units