document.write( "Question 1116970: For the hyperbola below, write the standard form equation. Then graph and tell the foci.
\n" ); document.write( "Vertices: (8, 14) and (8, -10); conjugate axis is 6 units long \n" ); document.write( "

Algebra.Com's Answer #731925 by KMST(5328) $\"\"$ $\"About$

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The transverse axis, connecting the vertices is vertical, and has a length of
\n" ); document.write( " $\"2a=14-%28-10%29%29=14%2B10=24\"$ , so $\"a=12\"$ .
\n" ); document.write( "The conjugate axis has a length of
\n" ); document.write( " $\"2b=6\"$ , so $\"b=3\"$ .
\n" ); document.write( "That makes the focal distance
\n" ); document.write( " $\"c=sqrt%28a%5E2%2Bb%5E2%29=sqrt%2812%5E2%2B3%5E2%29=sqrt%28144%2B9%29=sqrt%28154%29\"$ .
\n" ); document.write( "The center is the midpoint of the segment connecting the vertices,
\n" ); document.write( "so its coordinates are
\n" ); document.write( " $\"h=8\"$ and $\"k=%2814%2B%28-10%29%29%2F2=2\"$ .
\n" ); document.write( "Then, the coordinates of the foci are
\n" ); document.write( " $\"x=8\"$ (like the center and vertices), and
\n" ); document.write( " $\"y=2%2Bsqrt%28154%29\"$ and $\"y=2%2Bsqrt%28154%29\"$ .
\n" ); document.write( " $\"highlight%28F%5B1%5D=8%2C2%2Bsqrt%28154%29%29\"$ and $\"highlight%28F%5B2%5D=8%2C2-sqrt%28154%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Given the coordinates found for the center,
\n" ); document.write( "we know that the equation will involve
\n" ); document.write( "the squares of horizontal and vertical distances to the center:
\n" ); document.write( " $\"%28x-8%29%5E2\"$ and $\"%28y-2%29%5E2\"$ .
\n" ); document.write( "Obviously, as the vertices are a vertical distance $\"a=12\"$ from the center,
\n" ); document.write( " $\"%28y-2%29%5E2%3E=12%5E2\"$ <--> $\"%28y-2%29%5E2%2F12%5E2%3E=1\"$ ,
\n" ); document.write( "and the term $\"%28y-2%29%5E2%2F12%5E2%3E=1\"$ cannot be the one with the minus sign.
\n" ); document.write( "The equation is
\n" ); document.write( " $\"highlight%28%28y-2%29%5E2%2F12%5E2-%28x-8%29%5E2%2F3%5E2=1%29\"$ or $\"highlight%28%28y-2%29%5E2%2F144-%28x-8%29%5E2%2F9=1%29\"$ .
\n" ); document.write( "
\n" ); document.write( "GRAPH:
\n" ); document.write( "The two arms of the hyperbola are drawn in green and red.
\n" ); document.write( "Asymptotes, transverse and conjugate axes are in blue.
\n" ); document.write( "

The foci are the two black dots.
\n" ); document.write( "The foci are really close to the vertices,
\n" ); document.write( "but that is because the ratio of the lengths of transverse to conjugate axes is so large that $\"c\"$ is just a hair longer than $\"a\"$ . \n" ); document.write( "

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