Question

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

 4 (y − 1)² = − 7 (x − 3) 

Answer 1

 Given: 

4(y − 1)² = − 7 (x − 3) 

\[\Rightarrow \left( y - 1 \right)^2 = \frac{- 7}{4}\left( x - 3 \right)\] 

Let \[Y = y - 1\] 

\[X = x - 3\] 

Then, we have: 

\[Y^2 = \frac{- 7}{4}X\] 

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

\[4a = \frac{7}{4} \Rightarrow a = \frac{7}{16}\] 

∴ Vertex = (X = 0, Y = 0) = \[\left( x = 3, y = 1 \right)\] 

Focus = (X = −a, Y = 0) = \[\left( x - 3 = \frac{- 7}{16}, y - 1 = 0 \right) = \left( x = \frac{41}{16}, y = 1 \right)\] 

Equation of the directrix: 

X = a
i.e. \[x - 3 = \frac{7}{16} \Rightarrow x = \frac{55}{16}\] 

Axis = Y = 0
i.e. \[y - 1 = 0 \Rightarrow y = 1\] 

Length of the latus rectum = 4a = \[\frac{7}{4}\]