document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199700>Question 1199700</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The equation of hyper bola is ((X)^2/9)-((Y)^2/16)=1<BR>\n" );
document.write( "And it's asymptotes are y=4x/3 and y=-4x/3<BR>\n" );
document.write( "Here, h(x) represents the portion of the hyperbola in the first quadrant. Based on this:<BR>\n" );
document.write( "a. Write an expression for h(x) in terms of x .<BR>\n" );
document.write( "b. Use the expression from part a. to justify why h(x) lies below the line y=4x/3 . </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848198 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848198\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a. Find the expression for h(x)**\r<BR>\n" );
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document.write( "1. **Isolate y^2:**<BR>\n" );
document.write( "   - (x^2)/9 - (y^2)/16 = 1<BR>\n" );
document.write( "   - (y^2)/16 = (x^2)/9 - 1<BR>\n" );
document.write( "   - y^2 = 16 * [(x^2)/9 - 1]<BR>\n" );
document.write( "   - y^2 = (16x^2)/9 - 16\r<BR>\n" );
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document.write( "2. **Solve for y:**<BR>\n" );
document.write( "   - y = ±√[(16x^2)/9 - 16] \r<BR>\n" );
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document.write( "Since we're considering only the portion of the hyperbola in the first quadrant (where both x and y are positive), we take the positive square root:\r<BR>\n" );
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document.write( "   - h(x) = √[(16x^2)/9 - 16] \r<BR>\n" );
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document.write( "**b. Justify why h(x) lies below the line y = 4x/3**\r<BR>\n" );
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document.write( "1. **Consider the equation of the asymptote:**<BR>\n" );
document.write( "   - The asymptote in the first quadrant is y = 4x/3.\r<BR>\n" );
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document.write( "2. **Analyze the behavior of h(x):**<BR>\n" );
document.write( "   - h(x) represents the y-coordinate of the hyperbola for a given x-value.<BR>\n" );
document.write( "   - As x increases, the value of (16x^2)/9 increases significantly.<BR>\n" );
document.write( "   - However, the square root function grows relatively slowly compared to a linear function like y = 4x/3.\r<BR>\n" );
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document.write( "3. **Conclusion:**<BR>\n" );
document.write( "   - The square root in h(x) will cause the y-values of the hyperbola to increase at a slower rate than the linear increase of the asymptote y = 4x/3. <BR>\n" );
document.write( "   - Therefore, the graph of h(x) will always remain below the line y = 4x/3 in the first quadrant.\r<BR>\n" );
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document.write( "**In essence:**\r<BR>\n" );
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document.write( "* The hyperbola approaches the asymptote y = 4x/3 as x increases, but it never touches or crosses it.<BR>\n" );
document.write( "* The square root in the expression for h(x) limits the growth of the y-values compared to the linear growth of the asymptote.\r<BR>\n" );
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document.write( "This analysis demonstrates why the portion of the hyperbola in the first quadrant (h(x)) lies below the asymptote y = 4x/3.<BR>\n" );
document.write( " </SPAN>\n" );
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