Question

enter the equation of a parabola in the xy-plane that has a focus at (-10, -7) and a directrix of y = -3.

Explanation:

Step1: Recall the definition of a parabola

A parabola is the set of all points \((x, y)\) equidistant from the focus \((h, k + p)\) and the directrix \(y=k - p\). The standard form of a parabola that opens up or down is \((x - h)^2=4p(y - k)\), where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus (and also to the directrix).

Step2: Find the vertex \((h, k)\)

The vertex is halfway between the focus \((- 10,-7)\) and the directrix \(y = - 3\). The \(x\)-coordinate of the vertex is the same as the \(x\)-coordinate of the focus, so \(h=-10\). The \(y\)-coordinate of the vertex \(k\) is the average of the \(y\)-coordinate of the focus and the directrix value. So \(k=\frac{-7+( - 3)}{2}=\frac{-10}{2}=-5\).

Step3: Find the value of \(p\)

The distance \(p\) is the distance from the vertex \((-10,-5)\) to the focus \((-10,-7)\). Using the distance formula for vertical distance (since \(x\)-coordinates are the same), \(p=-7-( - 5)=-2\).

Step4: Substitute \(h\), \(k\) and \(p\) into the standard form

Substitute \(h = - 10\), \(k=-5\) and \(p=-2\) into the standard form \((x - h)^2=4p(y - k)\).
We get \((x+ 10)^2=4\times(-2)(y + 5)\)
Simplify the right - hand side: \(4\times(-2)=-8\), so the equation is \((x + 10)^2=-8(y + 5)\)

Answer:

\((x + 10)^2=-8(y + 5)\)