Question

write an equation of a parabola with vertex at the origin and the given focus.

1. focus at (-2, 0)
2. focus at (0, 4)
3. focus at (0, -3)
4. focus at (3, 0)
5. focus at (-5, 0)
6. focus at (0, 5)

Explanation:

Step1: Determine the form of the parabola

If the focus is \((a,0)\) (\(a
eq0\)), the parabola opens horizontally and its equation is \(y^{2} = 4ax\). If the focus is \((0,a)\) (\(a
eq0\)), the parabola opens vertically and its equation is \(x^{2}=4ay\).

Step2: Find the value of \(a\) for each focus

1. For focus \((- 2,0)\), \(a=-2\). The equation is \(y^{2}=4(-2)x=-8x\).
2. For focus \((0,4)\), \(a = 4\). The equation is \(x^{2}=4\times4y = 16y\).
3. For focus \((0,-3)\), \(a=-3\). The equation is \(x^{2}=4(-3)y=-12y\).
4. For focus \((3,0)\), \(a = 3\). The equation is \(y^{2}=4\times3x = 12x\).
5. For focus \((-5,0)\), \(a=-5\). The equation is \(y^{2}=4(-5)x=-20x\).
6. For focus \((0,5)\), \(a = 5\). The equation is \(x^{2}=4\times5y=20y\).

Answer:

1. \(y^{2}=-8x\)
2. \(x^{2}=16y\)
3. \(x^{2}=-12y\)
4. \(y^{2}=12x\)
5. \(y^{2}=-20x\)
6. \(x^{2}=20y\)