document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=207882>Question 207882</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the constant value of k so that the line having equation <BR>\n" );
document.write( "y =-3/4 x+k is tangent to the circle whose equation is (x-3)^2+(y+4)^2 = 25. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #157245 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=157245\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-10%2C10%2C-10%2C10%2C%28-3%2F4%29%2Ax%2B%289%2F2%29%2C%284%2F3%29%2Ax-8%2Csqrt%2825-%28x-3%29%5E2%29-4%2C-sqrt%2825-%28x-3%29%5E2%29-4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "your answer is k = 4.5<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "derivation of your answer is shown below:<BR>\n" );
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document.write( "the equation of the line tangent to the circle is y = -(3/4)*x + k<BR>\n" );
document.write( "we'll call that line P<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this line will have to be perpendicular to the radius of the circle at that point.<BR>\n" );
document.write( "we'll call that line R.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the slope of the line R must be equal to the negative reciprocal of the slope of the line P because the radius of the circle and the line tangent to it are perpendicular to each other at that point.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the slope of line P is -(3/4).<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the slope of line R must be (4/3) which is the negative reciprocal of (-3/4)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we know that the radius must go through the center of the circle.  This means that line R must go through the center of the circle.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we know that the center of the circle is (3,-4) because the formula of the circle tells us so.<BR>\n" );
document.write( "general formula of a circle is (x-h)^2 + (y-k)^2 = r^2 where (h,k) are the coordinates of the center of the circle.<BR>\n" );
document.write( "since -h = -3, then h = 3<BR>\n" );
document.write( "since -k = 4, then k = -4<BR>\n" );
document.write( "center of the circle is (3,-4)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "general form of the equation of line R is y = (4/3)*x + b<BR>\n" );
document.write( "where (4/3) is the slope and b is the y intercept.<BR>\n" );
document.write( "to find the y intercept of that line, we plug in a value for one of the coordinates of that line and solve for b.<BR>\n" );
document.write( "the center of the circle is one such point.<BR>\n" );
document.write( "general form of line R becomes:<BR>\n" );
document.write( "-4 = (4/3)*3 + b after we replace y with -4 and x with 3<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "solving for b, we get b = -8<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "equation of line R is y = (4/3)*x - 8<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we need to find the intersection of line R with the circle because that is also the point where line R will intersect with line P.<BR>\n" );
document.write( "once we find that point, we can then solve the equation for line P which will result in allowing us to find k.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we know the equation for line R.<BR>\n" );
document.write( "we know the equation for the circle.<BR>\n" );
document.write( "we can substitute the y value in the equation of the circle by the y value in the equation of line R.  this will allow us to find the x value of the intersection of the line R with the equation of the circle.<BR>\n" );
document.write( "once we find x, we can solve for y.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "here's how it was done.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the equation of the circle is (x-3)^2 + (y+4)^2 = 25<BR>\n" );
document.write( "the equation of line R is y = (4/3)*x - 8<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since (x-3)^2 equals x^2 - 6x + 9<BR>\n" );
document.write( "and since (y+4)^2 equals y^2 + 8y + 16<BR>\n" );
document.write( "then the equation of the circle expands to:<BR>\n" );
document.write( "x^2 - 6x + 9 + y^2 +8y + 16 = 25<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since we know that y = (4/3)*x - 8, we can substitute for y in this equation.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "8*y becomes 8*(4/3)*x - 64<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "y^2 becomes ((4/3)*x -8) * ((4/3)*x - 8) which becomes:<BR>\n" );
document.write( "(16/9)*x^2 - 8*(4/3)*x - 8*(4/3)*x + 64 which becomes:<BR>\n" );
document.write( "(16/9)*x^2 - 16*(4/3)*x + 64<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "y^2 + 8y becomes:<BR>\n" );
document.write( "(16/9)*x^2 - 16*(4/3)*x + 64 + 8*(4/3)*x - 64 which becomes:<BR>\n" );
document.write( "(16/9)*x^2 - 8*(4/3)*x<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the equation of the circle becomes:<BR>\n" );
document.write( "x^2 - 6x + 9 + (16/9)*x^2 - 8*(4/3)*x + 16 = 25<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "in order to remove the denominators, we multiply both sides of this equation by 9 to get:<BR>\n" );
document.write( "9x^2 - 54x + 81 + 16x^2 - 96x + 144 = 225<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "combining like terms we get:<BR>\n" );
document.write( "25x^2 - 150x + 225 = 225<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "subtracting 225 from both sides of this equation gets:<BR>\n" );
document.write( "25x^2 - 150x = 0<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "dividing both sides of this equation by 25 gets:<BR>\n" );
document.write( "x^2 - 6x = 0<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "factoring out the x gets:<BR>\n" );
document.write( "x * (x-6) = 0<BR>\n" );
document.write( "which results in:<BR>\n" );
document.write( "x = 0<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "x = 6<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "as shown on the graph, both these values for x are good.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "solving for y using line R and the value of x = 6 gets:<BR>\n" );
document.write( "y = (4/3)*6 - 8<BR>\n" );
document.write( "which results in:<BR>\n" );
document.write( "y = 0<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "solving for y using the original equation of the circle and the value of x = 6 gets:<BR>\n" );
document.write( "(6-3)^2 + (y+4)^2 = 25 which becomes:<BR>\n" );
document.write( "9 + y^2 + 8y + 16 = 25 which becomes:<BR>\n" );
document.write( "y^2 + 8y = 0 which becomes:<BR>\n" );
document.write( "y * (y-8) = 0 which becomes:<BR>\n" );
document.write( "y = 0<BR>\n" );
document.write( "or<BR>\n" );
document.write( "y = -8<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "when x = 6, y = 0 is good.<BR>\n" );
document.write( "when x = 0, y = -8 is good.<BR>\n" );
document.write( "this can be seen from the graph.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we now have the intersection of line R with the equation of the circle.<BR>\n" );
document.write( "we also have the intersection of line R with line P since both line R and line P intersect with the equation of the circle at that point.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "now we have a point for line P that we can use to solve that equation.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the equation for line P is y = (-3/4)*x + k<BR>\n" );
document.write( "if we plug in the value of 6 for x and the value of 0 for y, the equation becomes:<BR>\n" );
document.write( "0 = (-3/4)*(6) + k<BR>\n" );
document.write( "this results in:<BR>\n" );
document.write( "k = (3/4)*(6) which results in:<BR>\n" );
document.write( "k = 4.5<BR>\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );