Question

find an equation for the parabola with focus $(-2,0)$ and directrix $x = 2$.

Explanation:

Step1: Recall the definition of a parabola

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

Step2: Square both sides of the equation

Squaring both sides, we get \((x + 2)^2+y^2=(x - 2)^2\).
Expand the squares: \(x^{2}+4x + 4+y^{2}=x^{2}-4x + 4\).

Step3: Simplify the equation

Subtract \(x^{2}+4\) from both sides of the equation:
\(x^{2}+4x + 4+y^{2}-x^{2}-4=x^{2}-4x + 4-x^{2}-4\).
This simplifies to \(4x+y^{2}=-4x\).
Then, \(y^{2}=-8x\).

Answer:

\(y^{2}=-8x\)