Question

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1. set these two distances equal to each other.
2. simplify the resulting equation and express it in terms of ( x ) (for parabolas that open horizontally) or ( y ) (for parabolas that open vertically).

now, try this problem.
derive the equation of a parabola with a focus at ( (3,4) ) and a directrix at ( y = 2 ).
enter your answers in the boxes. be sure to include the correct signs.
( y=square(xsquare)^2square )

Explanation:

Step1: Recall the distance formula

The distance between a point \((x,y)\) and the focus \((3,4)\) is \(\sqrt{(x - 3)^2+(y - 4)^2}\). The distance between a point \((x,y)\) and the directrix \(y = 2\) is \(|y - 2|\).

Step2: Set the two distances equal

\(\sqrt{(x - 3)^2+(y - 4)^2}=|y - 2|\)

Step3: Square both sides

\((x - 3)^2+(y - 4)^2=(y - 2)^2\)

Step4: Expand the squares

\((x - 3)^2+y^{2}-8y + 16=y^{2}-4y+4\)

Step5: Simplify the equation

\((x - 3)^2+y^{2}-8y + 16-y^{2}+4y - 4 = 0\)
\((x - 3)^2-4y+12 = 0\)

Step6: Solve for \(y\)

\(4y=(x - 3)^2+12\)
\(y=\frac{1}{4}(x - 3)^2+3\)

Answer:

\(y=\frac{1}{4}(x - 3)^2+3\)