Question

What is the equation of the quadratic graph with a focus of (4,3) and a directrix of y=-6

Answer 1

Answer:

The correct option is B) $y=\frac{x^{2}}{6}-\frac{4x}{3}-\frac{11}{6}$

Step-by-step explanation:

We need to find out the quadratic graph with given focus $(4,-3)$ and a directrix $y=-6$

Using the distance formula, we find that the distance between (x,y) and the focus (4,-3) is $\sqrt{(x-4)^{2}+(y+3)^{2}}$ and the distance between (x,y) and the directrix y=-6, is $\sqrt{(y+6)^{2}}$

On the parabola, these distances are equal:

$\sqrt{(y+6)^{2}}=\sqrt{(x-4)^{2}+(y+3)^{2}}$

$(y+6)^{2}=(x-4)^{2}+(y+3)^{2}$

$y^{2}+36+12y=x^{2}+16-8x+y^{2}+9+6y$

Simplify the above

$36+12y-6y=x^{2}-8x+9+16$

$6y=x^{2}-8x+9+16-36$

$6y=x^{2}-8x-11$

Divide both the sides by 6,

$y=\frac{x^{2}}{6}-\frac{8x}{6}-\frac{11}{6}$

Simplified further,

$y=\frac{x^{2}}{6}-\frac{4x}{3}-\frac{11}{6}$

Therefore, the correct option is B) $y=\frac{x^{2}}{6}-\frac{4x}{3}-\frac{11}{6}$