Find the equation of the parabola given the focus (1, 4) and the directrix \(y = 1\). Then, sketch the graph.

ok, so for parabola a(x-h)^2 +k abs(a) = 1/(4c) where c is the distance between the focus and directrix.
The distance is 4-1, or 3 (y coord of the focus - y coord of directrix) so a = 1/(4*3) or 1/12 and the vertex (h,k) is the midpoint of the segment between the focus and directrix, which is (1, 2.5)
the parabola is 1/12*(x-1)^2 + 2.5

Answered by paulcox | 2024-06-10

The equation of the parabola with focus (1, 4) and directrix y = 1 is ( x − 1 ) 2 = 6 ( y − 2.5 ) . The vertex is located at (1, 2.5), and the parabola opens upwards. A sketch can be drawn with the vertex at (1, 2.5), focus at (1, 4), and directrix line at y = 1.
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Answered by paulcox | 2024-10-30