(y-7)^2 = 12x
Find the directrix
49/12
no matter. the proposed answer is bogus. The directrix is a line, not just a number.
Recall that the parabola
y^2 = 4px
has directrix at x = -p.
Your parabola is shifted up, but that does not change the directrix. So, the desired line is clearly
x = -3
see the properties at
http://www.wolframalpha.com/input/?i=parabola+%28y-7%29^2+%3D+12x
To find the directrix of the given equation, we need to rewrite it in a specific form called the vertex form. The vertex form of a parabola equation is (x-h)^2 = 4p(y-k), where (h, k) is the vertex and p is the distance from the vertex to the focus or the vertex to the directrix.
In the given equation, (y-7)^2 = 12x, we have the equation in terms of y. To get it into vertex form, we need to complete the square.
Step 1: Expand the equation.
y^2 - 14y + 49 = 12x
Step 2: Move the constant term (49) to the other side of the equation.
y^2 - 14y = 12x - 49
Step 3: Complete the square on the left side of the equation. To do this, take half of the coefficient of y (-14/2 = -7) and square it (-7^2 = 49). Add this value to both sides of the equation.
y^2 - 14y + 49 = 12x - 49 + 49
(y - 7)^2 = 12x
Now, the equation is in vertex form. The vertex is at the point (h, k), which is given by (0, 0) in this case.
Comparing this with the standard vertex form equation, we can see that p = 1/4. The directrix of the parabola is a horizontal line, located at a distance of p = 1/4 units below the vertex.
Therefore, the directrix is the line y = -1/4.