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Question
Find the vertex, focus, axis, directrix, and the length of the latus rectum of the parabola y2 – 8y – 8x + 24 = 0.
Solution
y2 – 8y – 8x + 24 = 0
⇒ y2 – 8y – 42 = 8x – 24 + 42
⇒ (y – 4)2 = 8x – 8
⇒ (y – 4)2 = 8(x – 1)
⇒ (y – 4)2 = 4(2) (x – 1)
∴ a = 2
Y2 = 4(2)X where X = x – 1 and Y = y – 4
| X, Y coordinates | x, y coordinates | |||
| Vertex (0, 0) | X = 0 | Y = 0 | x - 1 = 0 x = 1 |
y - 4 = 0 (1, 4) y = 4 |
| Focus (a, 0) | X = 2 | Y = 0 | x - 1 = 2 x = 2 + 1 = 3 |
y - 4 = 0 (3,4) y = 4 |
| Axis x-axis |
Y = 0 | y - 4 = 0 | y = 4 | |
| Directrix x + a = 0 |
X + 2 = 0 | x - 1 + 2 = 0 x + 1 = 0 |
x = - 1 | |
| Length of Latus rectum | 4a = 8 | |||
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