document.write( "Question 973502: Conics: Equation: 16y^2=49x^2+784\r
\n" ); document.write( "\n" ); document.write( "I need all necessary parts and need to know which conic it is \n" ); document.write( "

Algebra.Com's Answer #595729 by KMST(5328) $\"\"$ $\"About$

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$\"16y%5E2=49x%5E2%2B784\"$
\n" ); document.write( "The original equation can be transformed into an equivalent form
\n" ); document.write( "that will make all you need to know very clear.
\n" ); document.write( " $\"16y%5E2-49x%5E2=784\"$
\n" ); document.write( "It so happens that $\"16%2A49=784\"$ ,
\n" ); document.write( "one of those lucky coincidences that happen very often in math homework.
\n" ); document.write( " $\"%2816y%5E2-49x%5E2%29%2F784=784%2F784\"$
\n" ); document.write( " $\"16y%5E2%2F784-49x%5E2%2F784=1\"$
\n" ); document.write( " $\"y%5E2%2F49-x%5E2%2F16=1\"$
\n" ); document.write( " $\"y%5E2%2F7%5E2-x%5E2%2F4%5E2=1\"$
\n" ); document.write( "It is clear from the last equation above that for $\"x=0\"$ , $\"system%28y=-7%2C%22or%22%2Cy=7%29\"$ .
\n" ); document.write( "You can also see that since $\"x%5E2%2F4%5E2%3E%2B0\"$ ,
\n" ); document.write( "it must be true that $\"y%5E2%2F7%5E2%3E=1\"$ <---> $\"y%5E2%3E=7%5E2\"$ for all the points in the graph.
\n" ); document.write( "That means that the horizontal band with $\"-7%3Cy%3C7\"$ is a forbidden zone that the graph does not enter.
\n" ); document.write( "It also means that the points (0,-7), and (0,7) found above are the vertices, and that this is a hyperbola.
\n" ); document.write( " $\"y%5E2%2F7%5E2-x%5E2%2F4%5E2=1\"$ <---> $\"y%5E2%2F7%5E2=x%5E2%2F4%5E2%2B1\"$ <---> $\"y%5E2=7%5E2x%5E2%2F4%5E2%2B7%5E2\"$ <---> $\"y%5E2%2Fx%5E2=7%5E2%2F4%5E2%2B7%5E2%2Fx%5E2\"$ also makes it clear that as $\"x%5E2\"$ increases,
\n" ); document.write( "the curve approaches the lines where
\n" ); document.write( " $\"y%5E2%2Fx%5E2=7%5E2%2F4%5E2\"$ <---> $\"system%28y%2Fx=7%2F4%2C%22and%22%2Cy%2Fx=-7%2F4%29\"$ <---> $\"system%28y=%287%2F4%29x%2C%22and%22%2Cy=%28-7%2F4%29x%29\"$ .
\n" ); document.write( "Those two straight lines are the asymptotes.
\n" ); document.write( "So far we know that the curve looks like this
\n" ); document.write( "

,
\n" ); document.write( "a smile and a frown tangent on the oustside to the horizontal lines $\"system%28y=-7%2C%22and%22%2Cy=7%29\"$ ,
\n" ); document.write( "all symmetrical with respect to the x-axis and the origin, (0,0).
\n" ); document.write( "All that is left to do, is to draw that rectangular box bounded by the lines
\n" ); document.write( " $\"system%28y=-7%2C%22and%22%2Cy=7%29\"$ , and realize that the half sides, and half diagonals of that box area the $\"a\"$ , $\"b\"$ and $\"c\"$ that they showed you in formulas in math class,
\n" ); document.write( " $\"a\"$ , and $\"b\"$ being the numbers that appear squared as denominators in ,
\n" ); document.write( "and $\"c\"$ being the distance from the center, (0,0), to the foci, at (0,-c), and (0,c).
\n" ); document.write( "

In the right triangle, $\"c%5E2=7%5E2%2B4%5E2\"$ <--> $\"c%5E2=65\"$ --> $\"c=sqrt%2865%29\"$ \n" ); document.write( "

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