document.write( "Question 602433: The line mx-y=0 is tangent to the circle X2+Y2-6X+2y+2=0 find the two values of m. \n" ); document.write( "

Algebra.Com's Answer #380335 by KMST(5328) $\"\"$ $\"About$

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If the line $\"mx-y=0\"$ <--> $\"y=mx\"$ is tangent to the circle, they have only one point in common. So, substituting
\n" ); document.write( " $\"y=mx\"$ into the equation for the circle,
\n" ); document.write( " $\"x%5E2%2By%5E2-6x%2B7y%2B2=0\"$ , there should be only one point that is a solytion.
\n" ); document.write( "That would mean only one pair (x,y), meaning only one value for x.
\n" ); document.write( "So, here we go
\n" ); document.write( " $\"x%5E2%2B%28mx%29%5E2-6x%2B2%28mx%29%2B2=0\"$ --> $\"x%5E2%2Bm%5E2x%5E2-6x%2B2mx%2B2=0\"$ --> $\"%28m%5E2%2B1%29x%5E2%2B%282m-6%29x%2B2=0\"$
\n" ); document.write( "If a quadratic equation $\"ax%5E2%2Bbx%2Bc=0\"$ has only one solution its discriminant must be zero:
\n" ); document.write( " $\"b%5E2-4%2Aa%2Ac=0\"$
\n" ); document.write( "For the equation with m,
\n" ); document.write( " $\"a=m%5E2%2B1\"$ , $\"b=2m-6\"$ and $\"c=2\"$
\n" ); document.write( "The discriminant is
\n" ); document.write( " $\"%282m-6%29%5E2-4%2A%28m%5E2%2B1%29%2A2=0\"$ --> $\"4m%5E2-24m%2B36-8%28m%5E2%2B1%29=0\"$ --> $\"4m%5E2-24m%2B36-8m%5E2-8=0\"$ --> $\"-4m%5E2-24m%2B28=0\"$
\n" ); document.write( "Dividing both sides of the equal sign by (-4), we get as much friendlier equation:
\n" ); document.write( " $\"m%5E2%2B6m-7=0\"$
\n" ); document.write( "and factoring, we get
\n" ); document.write( " $\"%28m%2B7%29%28m-1%29=0\"$ with solutions $\"highlight%28m=1%29\"$ and $\"highlight%28m=-7%29\"$ .
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