Question

find an equation for the parabola with focus $(0,\frac{9}{2})$ and directrix $y = -\frac{9}{2}$

Explanation:

Step1: Recall the definition of a parabola

A parabola is the set of all points \((x,y)\) such that the distance from \((x,y)\) to the focus \((0,\frac{9}{2})\) is equal to the distance from \((x,y)\) to the directrix \(y =-\frac{9}{2}\).
The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). The distance from \((x,y)\) to the focus \((0,\frac{9}{2})\) is \(\sqrt{(x - 0)^2+(y-\frac{9}{2})^2}\), and the distance from \((x,y)\) to the directrix \(y=-\frac{9}{2}\) is \(\vert y+\frac{9}{2}\vert\).

Step2: Set up the equation

\(\sqrt{x^{2}+(y - \frac{9}{2})^{2}}=\vert y+\frac{9}{2}\vert\)
Square both sides: \(x^{2}+(y-\frac{9}{2})^{2}=(y + \frac{9}{2})^{2}\)
Expand using \((a\pm b)^{2}=a^{2}\pm2ab + b^{2}\):
\(x^{2}+y^{2}-9y+\frac{81}{4}=y^{2}+9y+\frac{81}{4}\)

Step3: Simplify the equation

Subtract \(y^{2}+\frac{81}{4}\) from both sides:
\(x^{2}-9y=9y\)
\(x^{2}=18y\)

Answer:

\(x^{2}=18y\)