Answer:
The correct option is B) [tex]y=\frac{x^{2}}{6}-\frac{4x}{3}-\frac{11}{6}[/tex]
Step-by-step explanation:
We need to find out the quadratic graph with given focus [tex](4,-3)[/tex] and a directrix [tex]y=-6[/tex]
Using the distance formula, we find that the distance between (x,y) and the focus (4,-3) is [tex]\sqrt{(x-4)^{2}+(y+3)^{2}}[/tex] and the distance between (x,y) and the directrix y=-6, is [tex]\sqrt{(y+6)^{2}}[/tex]
On the parabola, these distances are equal:
[tex]\sqrt{(y+6)^{2}}=\sqrt{(x-4)^{2}+(y+3)^{2}}[/tex]
[tex](y+6)^{2}=(x-4)^{2}+(y+3)^{2}[/tex]
[tex]y^{2}+36+12y=x^{2}+16-8x+y^{2}+9+6y[/tex]
Simplify the above
[tex]36+12y-6y=x^{2}-8x+9+16[/tex]
[tex]6y=x^{2}-8x+9+16-36[/tex]
[tex]6y=x^{2}-8x-11[/tex]
Divide both the sides by 6,
[tex]y=\frac{x^{2}}{6}-\frac{8x}{6}-\frac{11}{6}[/tex]
Simplified further,
[tex]y=\frac{x^{2}}{6}-\frac{4x}{3}-\frac{11}{6}[/tex]
Therefore, the correct option is B) [tex]y=\frac{x^{2}}{6}-\frac{4x}{3}-\frac{11}{6}[/tex]
