Question

enter the equation of a parabola in the xy-plane that has a focus at (-8, -5) and a directrix at the x-axis.

Explanation:

Step1: Recall the definition of a parabola

A parabola is the set of all points \((x, y)\) equidistant from the focus \((x_f, y_f)\) and the directrix. The distance from a point \((x, y)\) to the focus \((-8, -5)\) is \(\sqrt{(x + 8)^2+(y + 5)^2}\), and the distance from \((x, y)\) to the directrix \(y = 0\) (since the directrix is the \(x\)-axis) is \(|y - 0|=|y|\).

Step2: Set the distances equal

By the definition of a parabola, \(\sqrt{(x + 8)^2+(y + 5)^2}=|y|\). Square both sides to eliminate the square root and the absolute value: \((x + 8)^2+(y + 5)^2=y^2\).

Step3: Expand and simplify the equation

Expand \((y + 5)^2\): \((x + 8)^2+y^2+10y + 25=y^2\). Subtract \(y^2\) from both sides: \((x + 8)^2+10y+25 = 0\). Then solve for \(y\): \(10y=-(x + 8)^2 - 25\), so \(y=-\frac{1}{10}(x + 8)^2-\frac{25}{10}\), which simplifies to \(y=-\frac{1}{10}(x + 8)^2-\frac{5}{2}\).

Answer:

\(y =-\frac{1}{10}(x + 8)^2-\frac{5}{2}\)