document.write( "Question 1043331: Find the standard equation,foci,asymptotes and vertices of 25x^2-39y^2+150x+390y=-225 \n" ); document.write( "
Algebra.Com's Answer #658505 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
\"25x%5E2-39y%5E2%2B150x%2B390y=-225\"
\n" ); document.write( "\"25x%5E2%2B150x-39y%5E2%2B390y=-225\"
\n" ); document.write( "\"25%28x%5E2%2B6x%29-39%28y%5E2-10y%29=-225\"
\n" ); document.write( "\"25%28x%5E2%2B6x%2B9%29-39%28y%5E2-10y%2B25%29=-225%2B25%2A9-39%2A25\"
\n" ); document.write( "\"25%28x%5E2%2B6x%2B9%29-39%28y%5E2-10y%2B25%29=-225%2B225-975\"
\n" ); document.write( "\"25%28x%2B3%29%5E2-39%28y-5%29%5E2=-975\"
\n" ); document.write( "\"39%28y-5%29%5E2-25%28x%2B3%29%5E2=975\"
\n" ); document.write( "\"39%28y-5%29%5E2%2F975-25%28x%2B3%29%5E2%2F975=975%2F975\"
\n" ); document.write( "\"highlight%28%28y-5%29%5E2%2F25-%28x%2B3%29%5E2%2F39=1%29\"
\n" ); document.write( "The standard equation above tells us that
\n" ); document.write( "the center is at (-3,5) ;
\n" ); document.write( "the major (or transverse) axis is \"x=-3\";
\n" ); document.write( "the focal distance is \"c=sqrt%2825%2B39%29=sqrt%2864%29=8\" ;
\n" ); document.write( "the asymptotes have slopes such that \"slope%5E2=25%2F39\" , and
\n" ); document.write( "the y-coordinates of the vertices can be found from
\n" ); document.write( "\"%28y-5%29%5E2%2F25=1\" .
\n" ); document.write( "The asymptotes pass through center \"%22%28-3+%2C+5+%29%22\" ,
\n" ); document.write( "and have slopes such as
\n" ); document.write( "\"slope%5E2=25%2F39\"-->\"system%28slope=5sqrt%2839%29%2F39%2C%22or%22%2Cslope=-5sqrt%2839%29%2F39%29\" ,
\n" ); document.write( "the equations of the asymptotes (in point-slope form) are
\n" ); document.write( "\"highlight%28y-5=5sqrt%2839%29%28x%2B3%29%2F39%29\" and \"highlight%28y-5=-5sqrt%2839%29%28x%2B3%29%2F39%29\" .
\n" ); document.write( "Those equations can also be written as
\n" ); document.write( "\"highlight%28y=-5sqrt%2839%29x%2F39%2B%2865-15sqrt%2839%29%29%2F13%29\" and
\n" ); document.write( "\"highlight%28y=5sqrt%2839%29x%2F39%2B%2865%2B15sqrt%2839%29%29%2F13%29\" in slope-intercept form.
\n" ); document.write( "Since the foci are on the major axis,
\n" ); document.write( "at a distance \"c=8\" from the center,
\n" ); document.write( "their y- coordinates are \"y=5+%2B-+8\" , or \"system%28y=5%2B8=13%2C%22or%22%2Cy=5-8=-3%29\" .
\n" ); document.write( "So, one focus is at \"highlight%28%22%28+-3+%2C+13+%29%22%29\" ,
\n" ); document.write( "and the tother focus is at \"highlight%28%22%28+-3+%2C+-3+%29%22%29\"
\n" ); document.write( "As for the vertices, also on the major axis,
\n" ); document.write( "\"%28y-5%29%5E2%2F25=1\"-->\"%28y-5%29%5E2=25\"-->\"y=5+%2B-+sqrt%2825%29=+5+%2B-+5\"---> \"y=10%2C%22or%22%2Cy=0%29\" .
\n" ); document.write( "So the vertices are at \"highlight%28%22%28+-3+%2C+10+%29%22%29\" and \"highlight%28%22%28+-3+%2C+0+%29%22%29\" .
\n" ); document.write( "The hyperbola with major axis and foci looks like this:
\n" ); document.write( "
\n" ); document.write( " \n" ); document.write( "
\n" );