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Find an equation of the hyperbola having foci at (4, 8) and (4, 12) and vertices at (4,9) and (4, 11).

1 Answer

3 votes

Since the x-coordinate of the vertices are constant, we can conclude it is a vertical hyperbola. The equation for a vertical hyperbola is given by:


((y-k)^2)/(a^2)-((x-h)^2)/(b^2)=1

Where:


\begin{gathered} F=(h,k\pm c) \\ V=(h,k\pm a) \end{gathered}

so:


\begin{gathered} k+c=12 \\ so: \\ k=12-c \\ k-c=8 \\ 12-c-c=8 \\ 12-2c=8 \\ c=2 \\ k=10 \end{gathered}
\begin{gathered} k-a=9 \\ a=k-9 \\ a=1 \\ \end{gathered}

Therefore:


\begin{gathered} b=√(2^2-1^2) \\ b=√(3) \end{gathered}

So, the equation is:


((y-10)^2)/(1)-((x-4)^2)/(3)=1

Find an equation of the hyperbola having foci at (4, 8) and (4, 12) and vertices at-example-1
User GriFlo
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