document.write( "Question 1207940: The tangent line to a circle may be defined as the line that intersects the circle in a single point, called the point of tangency. If the equation of the circle is x^2 + y^2 = r^2 and the equation of the tangent line is y = mx + b, show that:\r
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\n" ); document.write( "\n" ); document.write( "A. r^2(1 + m^2) = b^2\r
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\n" ); document.write( "\n" ); document.write( "B. The point of tangency is [(-r^2 m)/b, (r^2/b)]\r
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\n" ); document.write( "\n" ); document.write( "C. The tangent line is perpendicular to the line containing the center of the circle and point of tangency. \r
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Algebra.Com's Answer #846099 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We find the x-intercept of y = mx + b, by setting y = 0.\r\n" );
document.write( "y = mx + b\r\n" );
document.write( "0 = mx + b\r\n" );
document.write( "-b = mx\r\n" );
document.write( "\"-b%2Fm=x\"\r\n" );
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document.write( "All 6 triangles ACO, AOB, OCB, ACO, ADC, DOC are similar, because a perpendicular\r\n" );
document.write( "drawn from the right angle to the hypotenuse divides a right triangle into\r\n" );
document.write( "two right triangles, each similar to it.\r\n" );
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document.write( "\"OC%5E%22%22%2FOA%5E%22%22=BC%5E%22%22%2FOB%5E%22%22\" \r\n" );
document.write( "\"r%5E%22%22%2F%28-b%2Fm%29=BC%5E%22%22%2Fb%5E%22%22\"\r\n" );
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document.write( "\"-mr%5E%22%22%2Fb%5E%22%22=BC%5E%22%22%2Fb%5E%22%22\"\r\n" );
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document.write( "\"BC=-mr\"\r\n" );
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document.write( "\"BC%5E2=m%5E2r%5E2\"\r\n" );
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document.write( "And by the Pythagorean theorem:\r\n" );
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document.write( "\"BC%5E2=OB%5E2-OC%5E2\"\r\n" );
document.write( "\"BC%5E2=b%5E2-r%5E2\"\r\n" );
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document.write( "So equating expressions for BC2\r\n" );
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document.write( "\"m%5E2r%5E2=b%5E2-r%5E2\"\r\n" );
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document.write( "\"r%5E2%2Bm%5E2r%5E2=b%5E2\"\r\n" );
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document.write( "\"r%5E2%281%2Bm%5E2%29=b%5E2\"\r\n" );
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document.write( "----------------------\r\n" );
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document.write( "For the coordinates of the point of tangency, C.  \r\n" );
document.write( "By similar triangles,\r\n" );
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document.write( "\"OD%2FOC=BC%2FOB\"\r\n" );
document.write( "\"OD%2Fr=%28-mr%29%2Fb\"\r\n" );
document.write( "\"OD=-mr%5E2%2Fb%5E%22%22\"  <--the x-coordinate of the point of tangency C\r\n" );
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document.write( "\"CD%2FOC=OC%2FOB\"\r\n" );
document.write( "\"OD%2Fr=r%2Fb\"\r\n" );
document.write( "\"OD=r%5E2%2Fb%5E%22%22\"  <--the y-coordinate of the point of tangency C\r\n" );
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document.write( "So the point of tangency C is \"%28matrix%281%2C3%2C-mr%5E2%2Fb%5E%22%22%2C%22%2C%22%2Cr%5E2%2Fb%5E%22%22%29%29\"\r\n" );
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document.write( "Edwin
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