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प्रश्न
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
y2 − 4y + 4x = 0
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उत्तर
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
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