document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=200407>Question 200407</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Analytic Geometry<BR>\n" );
document.write( "11. Determine the shortest distance from the origin to the line represented by y=1/2x-2.<BR>\n" );
document.write( "Can u please help  me ????????????<BR>\n" );
document.write( "thanksss </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #150685 by </B><A HREF=/tutors/aboutme.mpl?userid=vleith><B>vleith(2983)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=vleith><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=150685\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Draw the line.<BR>\n" );
document.write( "Then imagine a small circle centered at the origin. As that circle radius grows, the circle will eventually touch the line. The point where it 'just touches' results in the drawn line being a tangent to the circle.\r<BR>\n" );
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document.write( "What do you know about the radius of a circle at a tangent point and a tangent to a circle at that point?  They are perpendicular.\r<BR>\n" );
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document.write( "What do you know about the slopes of lines that are perpendicular? They are negative inverses. \r<BR>\n" );
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document.write( "So find the line that has a slope of -2 and contains the point (0,0).<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=find_line_by_slope_and_point.solver>FIND a line by slope and one point</A></TD></TR>\n" );
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document.write( "  What we know about the line whose equation we are trying to find out:<BR>\n" );
document.write( "    <UL><BR>\n" );
document.write( "      <LI>it goes through point (0, 0)</LI><BR>\n" );
document.write( "      <LI>it has a slope of -2</LI><BR>\n" );
document.write( "    </UL><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, let's draw a diagram of the coordinate system with point (0, 0) plotted with a little blue dot:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++++drawing%28+400%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A+++++++++++++grid%281%29%2C%0D%0A+++++++++++++blue%28+circle%28+0%2C+0%2C+0.2+%29+%29+%29%0D%0A++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <B>Write this down</B>: the formula for the equation, given point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D%2C+y%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D%2C+y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and intercept a, is<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax+%2B+%28y%5B1%5D-a%2Ax%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y=ax+%2B+%28y%5B1%5D-a%2Ax%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (see a paragraph below explaining why this formula is correct)<BR>\n" );
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document.write( "    Given that a=-2, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x%5B1%5D+=+0%2C+y%5B1%5D+=+0+%29+ ALIGN=MIDDLE  ALT=\"system%28+x%5B1%5D+=+0%2C+y%5B1%5D+=+0+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have the equation of the line:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2%2Ax+%2B+0 ALIGN=MIDDLE  ALT=\"y=-2%2Ax+%2B+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Explanation: <I>Why did we use formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax+%2B+%28y%5B1%5D+-+a%2Ax%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y=ax+%2B+%28y%5B1%5D+-+a%2Ax%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ? Explanation goes here. We are trying to find equation y=ax+b. The value of slope (a) is already given to us. We need to find b. If a point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >) lies on the line, it means that it satisfies the equation of the line. So, our equation holds for (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+=+a%2Ax%5B1%5D%2Bb ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+a%2Ax%5B1%5D%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > Here, we know a, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and do not know b. It is easy to find out: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=y%5B1%5D-a%2Ax%5B1%5D ALIGN=MIDDLE  ALT=\"b=y%5B1%5D-a%2Ax%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So, then, the equation of the line is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=ax%2B%28y%5B1%5D-a%2Ax%5B1%5D%29+ ALIGN=MIDDLE  ALT=\"+y=ax%2B%28y%5B1%5D-a%2Ax%5B1%5D%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </I><BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Here's the graph:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++++drawing%28+400%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A+++++++++++++grid%281%29%2C%0D%0A+++++++++++++graph%28+400%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C+-2%2Ax%2B0+%29%2C%0D%0A+++++++++++++blue%28+circle%28+0%2C+0%2C+0.2+%29+%29%2C%0D%0A+++++++++++++locate%28+0%2C+0%2C+%22%280%2C+0%29%22+%29+%29%0D%0A+++++++++++++%29%0D%0A++++++++++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "   </TD></TR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-2x ALIGN=MIDDLE  ALT=\"y+=+-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now that you have the equations for two lines, solve them to find the point they have in common.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%281%2F2%29x+-+2 ALIGN=MIDDLE  ALT=\"y+=+%281%2F2%29x+-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-2x ALIGN=MIDDLE  ALT=\"y+=+-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x+=+%281%2F2%29x+-+2 ALIGN=MIDDLE  ALT=\"-2x+=+%281%2F2%29x+-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%285%2F2%29x+=+-2 ALIGN=MIDDLE  ALT=\"-%285%2F2%29x+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x+=+4 ALIGN=MIDDLE  ALT=\"5x+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4%2F5 ALIGN=MIDDLE  ALT=\"x+=+4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-8%2F5 ALIGN=MIDDLE  ALT=\"y+=+-8%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now you have two points, the origin (0,0) and the point of intersection (0.8,-1.6). Find the length between them and you have your answer.<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=hummingbird_distance_2D_geo.solver>Distance between two points in two dimensions</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>The distance (denoted by d) between two points in two dimensions is given by the following formula:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Thus in our case, the required distance is<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%280-0.8%29%5E2+%2B+%280--1.6%29%5E2%29=+1.78885438199983+ ALIGN=MIDDLE  ALT=\"d=sqrt%28%280-0.8%29%5E2+%2B+%280--1.6%29%5E2%29=+1.78885438199983+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For more on this concept, refer to <A  rel=nofollow HREF=Distance_formula.wikipedia>Distance formula</A>.<BR>\n" );
document.write( "  </TD></TR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C+0.5x-2%2C+-2x%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C+0.5x-2%2C+-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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