Question

Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.Match the equation of a parabola to its focus and directrix

Answer 1

For the given question, we will find the equation of the parabola of each box

then, we will select the correct equation from the tiles

The first box: focus (2, -2) and directrix y = -8

So, the parabola will open up and the equation will be:

$\begin{gathered} h=2;k=\frac{-8+(-2)}{2}=-5 \\ (x-h)^2=4\cdot a(y-k) \\ a=3 \\ \\ (x-2)^2=12\cdot(y+5) \end{gathered}$

simplify the equation

$\begin{gathered} x^2-4x+4=12y+60 \\ y=\frac{x^2}{12}-\frac{x}{3}-\frac{14}{3} \end{gathered}$

The second box: Focus (-3, 6) and Directrix (x = -11)

So, the parabola will open right

The values of (a) and the vertex (h,k) will be:

$\begin{gathered} a=\frac{-3-(-11)}{2}=\frac{-3+11}{2}=\frac{8}{2}=4 \\ \\ h=\frac{-3+(-11)}{2}=-7;k=6 \end{gathered}$

The equation of the parabola will be:

$\begin{gathered} (y-k)^2=4a(x-h) \\ (y-6)^2=4\cdot4(x+7) \end{gathered}$

Simplifying the equation:

$\begin{gathered} y^2-12y+36=16x+112 \\ x=\frac{y^2}{16}-\frac{3y}{4}-\frac{19}{4} \end{gathered}$

The third box: Focus (2, -2); Directrix (x = 8)

So, the parabola will open left

The values of (a) and the vertex (h,k) will be:

$\begin{gathered} a=\frac{8-2}{2}=\frac{6}{2}=3 \\ h=\frac{8+2}{2}=5;k=-2 \end{gathered}$

The equation of the parabola will be:

$\begin{gathered} (y-k)^2=-4a(x-h) \\ (y+2)^2=-12(x-5) \\ \end{gathered}$

Simplifying the equation:

$\begin{gathered} y^2+4y+4=-12x+60 \\ x=-\frac{y^2}{12}-\frac{y}{3}+\frac{14}{3} \end{gathered}$

The fourth box: Focus (-7, 1) and Directrix (y = 11)

The parabola will open down

The values of (a) and the vertex (h,k) will be:

$\begin{gathered} a=\frac{11-1}{2}=\frac{10}{2}=5 \\ h=-7;k=\frac{11+1}{2}=\frac{12}{2}=6 \end{gathered}$

The equation of the parabola will be:

$\begin{gathered} (x-h)^2=-4a(y-k) \\ (x+7)^2=-20(y-6) \end{gathered}$

Simplifying the equation:

$\begin{gathered} x^2+14x+49=-20y+120 \\ \\ y=-\frac{x^2}{20}-\frac{7x}{10}+\frac{71}{20} \end{gathered}$

The drag of the tiles to the boxes according to the following figure: