document.write( "Question 1088731: given circle (x+3)^2+(y+2)^2=20 with tangent KL touching the circle at K.Determine
\n" ); document.write( "1 the length of KL,if point L=(7;-2)
\n" ); document.write( "2 the coordinates of point K,the point of contact of the tangent to the circle
\n" ); document.write( "3 the equations of the tangents KL \n" ); document.write( "

Algebra.Com's Answer #703033 by Fombitz(32388) $\"\"$ $\"About$

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KCL forms a right triangle (C is the center of the circle).
\n" ); document.write( "You know the hypotenuse (CL), you know the length of KC.
\n" ); document.write( "Solve for KL.
\n" ); document.write( " $\"CL%5E2=KC%5E2%2BKL%5E2\"$
\n" ); document.write( " $\"10%5E2=20%2BKL%5E2\"$
\n" ); document.write( " $\"KL%5E2=80\"$
\n" ); document.write( "So build a circle centered at (7,-2) with a radius squared of 80.
\n" ); document.write( " $\"+%28x-7%29%5E2%2B%28y%2B2%29%5E2=80+\"$
\n" ); document.write( "Find the intersection of the two circles, that'll be point K.
\n" ); document.write( "From the first circle,
\n" ); document.write( " $\"%28y%2B2%29%5E2=20-%28x%2B3%29%5E2\"$
\n" ); document.write( "Substitute into the second circle,
\n" ); document.write( " $\"%28x-7%29%5E2%2B20-%28x%2B3%29%5E2=80\"$
\n" ); document.write( " $\"x%5E2-14x%2B49-x%5E2-6x-9=60\"$
\n" ); document.write( " $\"-20x%2B40=60\"$
\n" ); document.write( " $\"-20x=20\"$
\n" ); document.write( " $\"x=-1\"$
\n" ); document.write( "Then,
\n" ); document.write( " $\"%28-1%2B3%29%5E2%2B%28y%2B2%29%5E2=20\"$
\n" ); document.write( " $\"4%2B%28y%2B2%29%5E2=20\"$
\n" ); document.write( " $\"%28y%2B2%29%5E2=16\"$
\n" ); document.write( " $\"y%2B2=0+%2B-+4\"$
\n" ); document.write( " $\"y=2\"$ or $\"y=-6\"$
\n" ); document.write( "Now that you know K (-1,2) and L (7,-2) you can get the line that goes through both.
\n" ); document.write( "Slope: $\"m=%28-2-2%29%2F%287-%28-1%29%29=-4%2F8=-1%2F2\"$ \r
\n" ); document.write( "\n" ); document.write( "Using the point slope form,
\n" ); document.write( " $\"+y-2=-%281%2F2%29%28x%2B1%29+\"$
\n" ); document.write( " $\"y-2=-x%2F2-1%2F2\"$
\n" ); document.write( " $\"y=-x%2F2%2B3%2F2\"$
\n" ); document.write( " $\"y=%283-x%29%2F2\"$ \r
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