document.write( "Question 758806: Find the equation of a circle passing through (1,2) and (3,4) and tangent to the line 3x+y=3. \n" ); document.write( "

Algebra.Com's Answer #461718 by MathLover1(20849) $\"\"$ $\"About$

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given:
\n" ); document.write( "the point A( $\"1\"$ , $\"2\"$ )
\n" ); document.write( "the point B( $\"3\"$ , $\"4\"$ )
\n" ); document.write( " $\"y=-3x%2B3\"$ \r
\n" ); document.write( "\n" ); document.write( "Plot A( $\"1\"$ , $\"2\"$ ) and B( $\"3\"$ , $\"4\"$ )in a rectangular coordinate system.
\n" ); document.write( "Draw the tangent line, $\"y=-3x%2B3\"$ , which cuts the x-axis at ( $\"1\"$ , $\"0\"$ ) .
\n" ); document.write( "By looking at the position of the points $\"A\"$ , $\"B\"$ , and at the path of the tangent
\n" ); document.write( "line, think that the point where the tangent touches the circle could be ( $\"1\"$ , $\"0\"$ )
\n" ); document.write( " or close to.\r
\n" ); document.write( "\n" ); document.write( "Plan: The center of the circle is at the intersection point of the perpendicular line to tangent at the point of tangency, and the perpendicular bisector of the segment $\"AB\"$ (reason: a tangent to a circle is perpendicular to its radius; the perpendicular bisector of a chord passes through the center).\r
\n" ); document.write( "\n" ); document.write( "Solution:\r
\n" ); document.write( "\n" ); document.write( "Let's suppose that ( $\"1\"$ , $\"0\"$ ) is the point of tangency.\r
\n" ); document.write( "\n" ); document.write( "Find the midpoint $\"M\"$ of $\"AB\"$ : \r
\n" ); document.write( "\n" ); document.write( " $\"M\"$ = ( $\"%281%2B3%29%2F2\"$ , $\"%282%2B4%29%2F2\"$ ) = ( $\"2\"$ , $\"+3\"$ )\r
\n" ); document.write( "\n" ); document.write( "Find the $\"slope\"$ of the line that passes though $\"A\"$ and $\"B\"$ : \r
\n" ); document.write( "\n" ); document.write( " $\"m+=+%284-2%29%2F3-1%29+=+1\"$ \r
\n" ); document.write( "\n" ); document.write( "Find the $\"equation\"$ of the $\"perpendicular\"$ bisector of $\"AB\"$ (whose slope must be $\"m%5B1%5D=-1\"$ and passes through ( $\"2\"$ , $\"+3\"$ )):\r
\n" ); document.write( "\n" ); document.write( " $\"y+=+-1x+%2B+b\"$ ; ....let ( $\"x\"$ , $\"+y\"$ )= ( $\"2\"$ , $\"+3\"$ )\r
\n" ); document.write( "\n" ); document.write( " $\"3+=+-1%2A2+%2B+b\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"b+=+5\"$ \r
\n" ); document.write( "\n" ); document.write( "so, the $\"equation\"$ of the $\"perpendicular\"$ bisector of $\"AB\"$ is\r
\n" ); document.write( "\n" ); document.write( " $\"y+=+-x+%2B+5\"$ \r
\n" ); document.write( "\n" ); document.write( "Find the equation of the perpendicular line to the tangent (whose slope is $\"1%2F3\"$ since the perpendicular lines have negative reciprocal slopes) and passes through ( $\"1\"$ , $\"0\"$ )):\r
\n" ); document.write( "\n" ); document.write( " $\"y+=+%281%2F3%29x+%2B+b\"$ ;...... let ( $\"x\"$ , $\"y\"$ )= ( $\"1\"$ , $\"0\"$ )\r
\n" ); document.write( "\n" ); document.write( " $\"0+=+%281%2F3%29%2A1+%2B+b\"$
\n" ); document.write( " $\"b+=+-1%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"y+=+%281%2F3%29x+-+1%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( "Find the intersection point of $\"y+=+-x+%2B+5\"$ and $\"y+=+%281%2F3%29x+-+1%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"-x+%2B+5+=+%281%2F3%29x+-+1%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"%284%2F3%29x+=+16%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+16%2F3+%2A+3%2F4\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+4\"$ implies $\"y+=1\"$ \r
\n" ); document.write( "\n" ); document.write( "So the point ( $\"4\"$ , $\"+1\"$ )... ..=> could be the $\"center\"$ of the circle.\r
\n" ); document.write( "\n" ); document.write( "If we compute the distances between ( $\"4\"$ , $\"+1\"$ ) and ( $\"1\"$ , $\"0\"$ ), ( $\"1\"$ , $\"2\"$ ), ( $\"3\"$ , $\"4\"$ ), we see that they have the same length, $\"sqrt%2810%29\"$ , then we say the center of the circle is ( $\"4\"$ , $\"1\"$ ) and the radius has a length of $\"sqrt%2810%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "Thus, the equation of the circle that passes through ( $\"1\"$ , $\"2\"$ ) and ( $\"3\"$ , $\"4\"$ ) is the equation of a circle with center ( $\"4\"$ , $\"1\"$ ) and a radius of $\"sqrt%2810%29\"$ .\r
\n" ); document.write( "\n" ); document.write( " $\"%28x+-+4%29%5E2+%2B+%28y+-+1%29%5E2+=+10\"$ \r
\n" ); document.write( "\n" ); document.write( "Let's solve the problem without risking extra work (we clearly see that our guess for the point of tangency was right).\r
\n" ); document.write( "\n" ); document.write( "After we draw the given information on the rectangular system, we would like to draw a parallel line to the tangent line that passes through the point B ( $\"3\"$ , $\"4\"$ ).
\n" ); document.write( "For this we need the equation of the line to use its slope to draw it and to find the point of intersection with the perpendicular bisector of segment $\"AB\"$ (since the center of the circle lies there; see above); (the slope is $\"-3\"$ , since parallel line have the same slope).\r
\n" ); document.write( "\n" ); document.write( " $\"y+=+-3x+%2B+b\"$ ; .......let ( $\"x\"$ , $\"y\"$ ) = ( $\"3\"$ , $\"4\"$ )\r
\n" ); document.write( "\n" ); document.write( " $\"4+=+-3%2A3+%2B+b\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"b+=+13\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"y+=+-3x+%2B+13\"$ \r
\n" ); document.write( "\n" ); document.write( "Let's find the intersection point of $\"y+=+-x+%2B+5\"$ and $\"y+=+-3x+%2B+13\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"-x+%2B+5+=+-3x+%2B+13\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"2x+=+8\"$
\n" ); document.write( " $\"x+=+4\"$ implies $\"y+=+1\"$ , say the center $\"O\"$ is at ( $\"4\"$ , $\"1\"$ )\r
\n" ); document.write( "\n" ); document.write( "Let's find the length of the congruent ( $\"O\"$ lies on the $\"perpendicular\"$ bisector) segments $\"OA\"$ and $\"OB\"$ , maybe they are radii.\r
\n" ); document.write( "\n" ); document.write( " $\"OA+=+OB+=+sqrt%28%284-1%292%2B%281-2%292%29+=+sqrt%2810%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "Since the point $\"O\"$ lies in the same time on the perpendicular bisector of $\"AB\"$ and on the parallel line to the tangent, let's find the equation of the perpendicular line to the tangent that passes through $\"O\"$ (the slope is $\"1%2F3\"$ since the perpendicular lines have negative reciprocal slopes):\r
\n" ); document.write( "\n" ); document.write( " $\"y+=+%281%2F3%29x+%2B+b\"$ ;.... let ( $\"x\"$ , $\"y\"$ ) = ( $\"4\"$ , $\"1\"$ )\r
\n" ); document.write( "\n" ); document.write( " $\"1+=+%281%2F3%29%2A4+%2B+b\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"b+=+-1%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"y+=+%281%2F3%29x+-+1%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( "Let's find the intersection point of $\"y+=+-3x+%2B+3\"$ and $\"y+=+%281%2F3%29x+-+1%2F3\"$ (maybe this would be the point of tangency):\r
\n" ); document.write( "\n" ); document.write( " $\"-3x+%2B+3+=+%281%2F3%29x+-+1%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"%284%2F3%29x+=+4%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"x+=+1\"$ implies $\"y+=+0\"$ ; say T( $\"1\"$ , $\"0\"$ )\r
\n" ); document.write( "\n" ); document.write( "Let's find the length of $\"OT\"$ :\r
\n" ); document.write( "\n" ); document.write( " $\"OT+=+sqrt%28%284-1%292%2B%281-0%292%29+=+sqrt%2810%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "Since $\"OA\"$ , $\"OB\"$ , and $\"OT\"$ have the same length, they are radii of the circle with center O( $\"4\"$ , $\"1\"$ ). So our guess was right!\r
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