document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1132381>Question 1132381</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The straight line 3x-4y-3=0 cuts the circle x^2+y^2-4x-2y+4=0. Find (i)the coordinates of its centre (ii)the radius (iii)equations of tangent and normal to this circle from the origin </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #749400 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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/cgi-bin/show-face.mpl?user=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=749400\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The straight line 3x-4y-3=0 cuts the circle x^2+y^2-4x-2y+4=0. Find (i)the coordinates of its centre (ii)the radius (iii)equations of tangent and normal to this circle from the origin<BR>\n" );
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document.write( "Find (i)the coordinates of its centre<BR>\n" );
document.write( "x^2+y^2-4x-2y+4=0<BR>\n" );
document.write( "x^2+y^2-4x-2y = -4<BR>\n" );
document.write( "x^2-4x + y^2-2y = -4<BR>\n" );
document.write( "Complete the squares for x and y:<BR>\n" );
document.write( "x^2-4x+4 + y^2-2y+1 = -4+4+1 = 1<BR>\n" );
document.write( "(x-2)^2 + (y-1)^2 = 1<BR>\n" );
document.write( "Center at (2,1)<BR>\n" );
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document.write( "(ii)the radius<BR>\n" );
document.write( "r = 1<BR>\n" );
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document.write( "(iii)equations of tangent and normal to this circle from the origin<BR>\n" );
document.write( "One tangent is the x-axis, or y = 0.<BR>\n" );
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document.write( "The distance from the Origin to (2,1) is sqrt(5).<BR>\n" );
document.write( "Label the Origin O, the center C, the tangent point P, and the point (2,0) T.<BR>\n" );
document.write( "Angle CPO is 90 degs.<BR>\n" );
document.write( "OC = sqrt(5)<BR>\n" );
document.write( "CP = 1, the radius of the circle.<BR>\n" );
document.write( "OP = 2<BR>\n" );
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document.write( "Slope of OC = 1/2 = tangent of the angle between the x-axis and OC<BR>\n" );
document.write( "Triangles OCP and OCT are congruent --> angle COT = and COP = arctan(1/2)<BR>\n" );
document.write( "Slope of OP = tan(2*tan(POC)) = 4/3<BR>\n" );
document.write( "y = 4x/3 is the 2nd tangent line.<BR>\n" );
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document.write( "Not clear what you mean by the lines normal, and I don't see how the given line is relevant.\r<BR>\n" );
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