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Find the focus, directrix, and focal diameter of the parabola. 2xsqaured +13y = 0

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1 Answer

This equation uses x^2=4py to find the focus, where (0,p) is the focus. Since x^2 equals -13y (after subtracting 13y from both sides of the equation), this means that -13y=4py -> -13=4p -> p=-13/4.
 
So we know the focus is (0,-13/4). Since the vertex of the parabola is always halfway between the focus and the directrix, and we know the vertex of the parabola is at (0,0), we know that the directrix must pass through the point (0,0-(-13/4)) (subtract the y value of the focus from the y value of the vertex, or you can simply take the opposite of the focal point when the vertex is at (0,0)), giving us the point (0,13/4). Since the directrix remains parallel to the x axis for this equation, the directrix is located at y=13/4. The focal diameter is |4p|, or 13.
 
In conclusion:
The focus is (0,-13/4)
The directrix is y=13/4
The focal diameter is 13

Comments

2x2+13y=0 becomes x2=(-13/2)y.
Therefore, wouldn't 4p=-13/2, and p=-13/8?
Yes sorry. I see people post x2 instead of x^2 so I got confused seeing 2xsquared and did not divide by 2.