document.write( "Question 669435: find an equation of a circle that tangent is to 3x+4y-15=0 at (5,0); contains (-2,-1) \n" ); document.write( "

Algebra.Com's Answer #416409 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
Maybe there is a formula in your book that solves this type of problem.
\n" ); document.write( "I do not do types or formulas, so my solution will just stem from the most basic concepts. Sorry if it gets long and boring.
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\n" ); document.write( "CIRCLE TANGENT TO A LINE:
\n" ); document.write( "At the point where a circle is tangent to a line, the radius is perpendicular to that line.
\n" ); document.write( "So the center of that circle is on the line perpendicular to $\"3x%2B4y-15=0\"$ at point (5,0).
\n" ); document.write( " $\"3x%2B4y-15=0\"$ <--> $\"3x%2B4y=15\"$ <--> $\"4y=-3x%2B15\"$ <--> $\"y=%28-3%2F4%29x%2B15%2F4\"$ ,
\n" ); document.write( "so the line $\"3x%2B4y-15=0\"$ has a slope of $\"-3%2F4\"$
\n" ); document.write( "The slope of a line perpendicular to $\"3x%2B4y-15=0\"$ is
\n" ); document.write( " $\"%28-1%29%2F%28%28-3%2F4%29%29\"$ = $\"4%2F3\"$
\n" ); document.write( "The equation of a line with slope $\"4%2F3\"$ that passes through (5,0) is
\n" ); document.write( " $\"y-0=%284%2F3%29%28x-5%29\"$ <--> $\"highlight%28y=%284%2F3%29x-20%2F3%29\"$
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\n" ); document.write( "CIRCLE PASSING THROUGH TWO POINTS:
\n" ); document.write( "The center of the circle that passes through (5,0) and (-2,-1) is at the same distance from those two points,
\n" ); document.write( "so it is on the perpendicular bisector of the segment joining those two points.
\n" ); document.write( "The perpendicular bisector is perpendicular to that segment and passes through the midpoint of that segment.
\n" ); document.write( "The midpoint of that segment, calculated from the coordinates of (5,0) and (-2,-1) is
\n" ); document.write( "( $\"%285%2B%28-2%29%29%2F2\"$ , $\"%280%2B%28-1%29%29%2F2\"$ )=( $\"3%2F2\"$ , $\"-1%2F2\"$ )
\n" ); document.write( "The slope of that segment, calculated from the coordinates of (5,0) and (-2,-1) is
\n" ); document.write( " $\"%280-%28-1%29%29%2F%285-%28-2%29%29=1%2F7\"$
\n" ); document.write( "and the slope of a perpendicular line is $\"%28-1%29%2F%281%2F7%29\"$ = $\"-7\"$
\n" ); document.write( "The equation of the line with slope $\"-7\"$ passing through ( $\"3%2F2\"$ , $\"-1%2F2\"$ ) is
\n" ); document.write( " $\"y-%28-1%2F2%29=-7%28x-3%2F2%29\"$ <--> $\"y%2B1%2F2=-7x%2B21%2F2\"$ <--> $\"y=-7x%2B21%2F2-1%2F2\"$ <--> $\"y=-7x%2B20%2F2\"$ <--> $\"highlight%28y=-7x%2B10%29\"$
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\n" ); document.write( "THE CENTER OF OUR CIRCLE:
\n" ); document.write( "The center of the circle we are looking for is on the lines
\n" ); document.write( " $\"y=%284%2F3%29x-20%2F3\"$ and $\"y=-7x%2B10\"$ .
\n" ); document.write( "Those lines are not parallel, so they intersect at one point.
\n" ); document.write( "That point is the center of the circle.
\n" ); document.write( "We need to solve $\"system%28y=%284%2F3%29x-20%2F3%2Cy=-7x%2B10%29\"$
\n" ); document.write( " $\"%284%2F3%29x-20%2F3=-7x%2B10\"$ --> $\"4x-20=-21x%2B30\"$ (multiplying both sides times 3)
\n" ); document.write( " $\"4x-20=-21x%2B30\"$ --> $\"4x=-21x%2B30%2B20\"$ --> $\"4x=-21x%2B50\"$ --> $\"4x%2B21x=50\"$ --> $\"25x=50\"$ --> $\"x=50%2F25\"$ --> $\"highlight%28x=2%29\"$
\n" ); document.write( "Substituting $\"x=2\"$ into $\"y=-7x%2B10\"$ ,
\n" ); document.write( " $\"y=-7%2A2%2B10\"$ --> $\"y=-14%2B10\"$ --> $\"highlight%28y=-4%29\"$
\n" ); document.write( "So the center of the circle is (2,-4).
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\n" ); document.write( "THE RADIUS OF THE CIRCLE:
\n" ); document.write( "The radius $\"r\"$ is the distance from circle center (2,-4) to circle point (5,0).
\n" ); document.write( "We can calculate $\"r%5E2=%285-2%29%5E2%2B%280-%28-4%29%29%5E2=3%5E2%2B%28-4%29%5E2=9%2B16=highlight%2825%29\"$
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\n" ); document.write( "THE EQUATION OF THE CIRCLE\"
\n" ); document.write( "The equation of a circle centered at (2,-4) and with $\"r%5E2=25\"$ is
\n" ); document.write( " $\"%28x%2B2%29%5E2%2B%28y-%28-4%29%29%5E2=25\"$ --> $\"highlight%28%28x%2B2%29%5E2%2B%28y%2B4%29%5E2=25%29\"$ \n" ); document.write( "

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