Question
Find the directrix of the parabola y^2=4(x+2y)
Answers
GPT 3.5
To find the directrix of the parabola \(y^2 = 4(x + 2y)\), we first need to rewrite the equation in standard form, which is \(4p(y-k) = (x-h)^2\) for a parabola that opens to the right or left.
Rearranging the given equation:
\[y^2 = 4(x + 2y)\]
\[y^2 = 4x + 8y\]
\[y^2 - 8y = 4x\]
\[y^2 - 8y + 16 = 4x + 16\]
\[(y-4)^2 = 4(x + 4)\]
From the standard form equation above, we see that the vertex of the parabola is at point \((h, k) = (-4, 4)\), and \(4p = 4\), so \(p = 1\).
For a parabola that opens to the right, we know that the directrix is a vertical line p units to the left of the vertex. In this case, the equation of the directrix is:
\[x = -4 - p\]
\[x = -4 - 1\]
\[x = -5\]
Therefore, the directrix of the parabola \(y^2 = 4(x + 2y)\) is the vertical line x = -5.
Rearranging the given equation:
\[y^2 = 4(x + 2y)\]
\[y^2 = 4x + 8y\]
\[y^2 - 8y = 4x\]
\[y^2 - 8y + 16 = 4x + 16\]
\[(y-4)^2 = 4(x + 4)\]
From the standard form equation above, we see that the vertex of the parabola is at point \((h, k) = (-4, 4)\), and \(4p = 4\), so \(p = 1\).
For a parabola that opens to the right, we know that the directrix is a vertical line p units to the left of the vertex. In this case, the equation of the directrix is:
\[x = -4 - p\]
\[x = -4 - 1\]
\[x = -5\]
Therefore, the directrix of the parabola \(y^2 = 4(x + 2y)\) is the vertical line x = -5.