Question

what is the equation of a parabola with a focus at (2, -3) and a directrix of x = 5, as shown?
a ( y^{2}=-4x - 4y + 12 )
b ( y^{2}=-6x - 4y + 12 )
c ( y^{2}=4x - 6y + 12 )
d ( y^{2}=-6x - 6y + 12 )

Explanation:

Step1: Recall the definition of a parabola

The distance from any point \((x,y)\) on the parabola to the focus \((2,-3)\) is equal to the distance from the point \((x,y)\) to the directrix \(x = 5\).
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 \(|x - a|\).
So, \(\sqrt{(x - 2)^2+(y+ 3)^2}=|x - 5|\).

Step2: Square both sides of the equation

\((x - 2)^2+(y + 3)^2=(x - 5)^2\).
Expand the squares: \(x^{2}-4x + 4+y^{2}+6y+9=x^{2}-10x + 25\).

Step3: Simplify the equation

Cancel out \(x^{2}\) on both sides: \(-4x + 4+y^{2}+6y+9=-10x + 25\).
Rearrange the terms: \(y^{2}=-6x-6y + 12\).

Answer:

D. \(y^{2}=-6x - 6y+12\)