Question

lauren describes a parabola where the focus has a positive, nonzero x coordinate. which parabola(s) could lauren be describing? check all that apply.
□ (x^{2}=4y)
□ (x^{2}=-6y)
□ (y^{2}=x)
□ (y^{2}=10x)
□ (y^{2}=-3x)
□ (y^{2}=5x)

Explanation:

Step1: Recall focus - formula for parabolas

For parabola of the form $x^{2}=4py$, the focus is $(0,p)$; for parabola of the form $y^{2}=4px$, the focus is $(p,0)$.

Step2: Analyze $x^{2}=4y$

Here $4p = 4\Rightarrow p = 1$, focus is $(0,1)$, $x -$coordinate is 0.

Step3: Analyze $x^{2}=-6y$

Here $4p=-6\Rightarrow p =-\frac{3}{2}$, focus is $(0,-\frac{3}{2})$, $x -$coordinate is 0.

Step4: Analyze $y^{2}=x$

Here $4p = 1\Rightarrow p=\frac{1}{4}$, focus is $(\frac{1}{4},0)$, $x -$coordinate is positive.

Step5: Analyze $y^{2}=10x$

Here $4p = 10\Rightarrow p=\frac{5}{2}$, focus is $(\frac{5}{2},0)$, $x -$coordinate is positive.

Step6: Analyze $y^{2}=-3x$

Here $4p=-3\Rightarrow p =-\frac{3}{4}$, focus is $(-\frac{3}{4},0)$, $x -$coordinate is negative.

Step7: Analyze $y^{2}=5x$

Here $4p = 5\Rightarrow p=\frac{5}{4}$, focus is $(\frac{5}{4},0)$, $x -$coordinate is positive.

Answer:

$y^{2}=x$, $y^{2}=10x$, $y^{2}=5x$