We are asked to find the standard equation of a parabola given its focus or directrix, assuming that the vertex is at the origin. We need to solve problems 9, 10, 11, 12, 13, and 14.

1. Problem Description

We are asked to find the standard equation of a parabola given its focus or directrix, assuming that the vertex is at the origin. We need to solve problems 9, 10, 11, 12, 13, and

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2. Solution Steps

Problem 9: Focus is at $(2,0)$.

Since the focus is at $(2,0)$, the parabola opens to the right, and the equation is of the form $y^2 = 4px$, where the focus is at $(p,0)$.

Here, $p=2$. Therefore, the equation is $y^2 = 4(2)x$, which simplifies to $y^2 = 8x$.

Problem 10: Directrix is $x=3$.

Since the directrix is $x=3$, the parabola opens to the left, and the equation is of the form $y^2 = 4px$, where the directrix is $x=-p$.

Here, $-p=3$, so $p=-3$. Therefore, the equation is $y^2 = 4(-3)x$, which simplifies to $y^2 = -12x$.

Problem 11: Directrix is $y-2=0$, or $y=2$.

Since the directrix is $y=2$, the parabola opens downwards, and the equation is of the form $x^2 = 4py$, where the directrix is $y=-p$.

Here, $-p=2$, so $p=-2$. Therefore, the equation is $x^2 = 4(-2)y$, which simplifies to $x^2 = -8y$.

Problem 12: Focus is $(0, -\frac{1}{9})$.

Since the focus is at $(0, -\frac{1}{9})$, the parabola opens downwards, and the equation is of the form $x^2 = 4py$, where the focus is at $(0,p)$.

Here, $p=-\frac{1}{9}$. Therefore, the equation is $x^2 = 4(-\frac{1}{9})y$, which simplifies to $x^2 = -\frac{4}{9}y$.

Problem 13: Focus is $(-4,0)$.

Since the focus is at $(-4,0)$, the parabola opens to the left, and the equation is of the form $y^2 = 4px$, where the focus is at $(p,0)$.

Here, $p=-4$. Therefore, the equation is $y^2 = 4(-4)x$, which simplifies to $y^2 = -16x$.

Problem 14: Directrix is $y = \frac{7}{2}$.

Since the directrix is $y=\frac{7}{2}$, the parabola opens downwards, and the equation is of the form $x^2 = 4py$, where the directrix is $y=-p$.

Here, $-p=\frac{7}{2}$, so $p=-\frac{7}{2}$. Therefore, the equation is $x^2 = 4(-\frac{7}{2})y$, which simplifies to $x^2 = -14y$.

3. Final Answer

9. $y^2 = 8x$

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0. $y^2 = -12x$

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1. $x^2 = -8y$

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2. $x^2 = -\frac{4}{9}y$

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3. $y^2 = -16x$

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4. $x^2 = -14y$