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प्रश्न
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
y2 = 8x + 8y
उत्तर
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
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