Question

Write the equation for a parabola with a focus at (-4,-2) and a directrix at y = 3.

Answer 1

Answer:

$\displaystyle y=-\frac{1}{10}x^2-\frac{4}{5}x-\frac{11}{10}$

Step-by-step explanation:

By definition, any point (x, y) on the parabola is equidistant from the focus and the directrix.

The distance between a point (x, y) on the parabola and the focus can be described using the distance formula:

$d=\sqrt{(x-(-4))^2+(y-(-2))^2$

Simplify:

$d=\sqrt{(x+4)^2+(y+2)^2}$

Since the directrix is an equation of y, we will use the y-coordinate. The vertical distance between a point (x, y) on the parabola and the directrix can be described using absolute value:

$d=|y-3|\text{ or } |3-y|$

The two equations are equivalent. Therefore:

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

Solve for y. We can square both sides. Since anything squared is positive, we can remove the absolute value:

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

Expand:

$(x^2+8x+16)+(y^2+4y+4)=(y^2-6y+9)$

Isolate:

$x^2+8x+11=-10y$

Divide both sides by -10. Hence, our equation is:

$\displaystyle y=-\frac{1}{10}x^2-\frac{4}{5}x-\frac{11}{10}$