document.write( "Question 1009850: Find the equation of the line a) parallel b) perpendicular to 2x-3y=6 and passing at a distance 2sqrt (3) units from (-1,2) \n" ); document.write( "

Algebra.Com's Answer #625358 by KMST(5328) $\"\"$ $\"About$

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THE EXPECTED WAY TO SOLVE IT:
\n" ); document.write( "You were taught the the distance from a point $\"P%28x%5B0%5D%2Cy%5B0%5D%29\"$
\n" ); document.write( "to a line $\"ax%2Bby%2Bc=0\"$ is measured along segment $\"PQ\"$ ,
\n" ); document.write( "from $\"P\"$ to the closest point on the line, point $\"Q\"$ .
\n" ); document.write( "They also taught you a formula to find that distance:
\n" ); document.write( "

,
\n" ); document.write( "and maybe even formulas to find the coordinates of point $\"Q\"$ .
\n" ); document.write( "a) Lines parallel to $\"2x-3y=6\"$ <--> $\"2x-3y-6=0\"$
\n" ); document.write( "Have the equation $\"2x-3y%2Bc=0\"$ .
\n" ); document.write( "Blindly applying that formula,
\n" ); document.write( "

\n" ); document.write( " $\"d%282x-3y%2Bc=0%2CP%28-1%2C2%29%29=abs%28-2-6%2Bc%29%2Fsqrt%284%2B9%29\"$
\n" ); document.write( " $\"d%282x-3y%2Bc=0%2CP%28-1%2C2%29%29=abs%28c-8%29%2Fsqrt%2813%29\"$ .
\n" ); document.write( "And since we know that distance must be $\"2sqrt%283%29\"$ ,
\n" ); document.write( " $\"abs%28c-8%29%2Fsqrt%2813%29=2sqrt%283%29\"$ --> $\"abs%28c-8%29=2sqrt%283%29%2Asqrt%2813%29\"$ --> $\"abs%28c-8%29=2sqrt%2839%29\"$ --> $\"c=8+%2B-+2sqrt%2839%29\"$ .
\n" ); document.write( "So the equation of the two lines parallel to $\"2x-3y=6\"$ ,
\n" ); document.write( "and passing at a distance $\"2sqrt%283%29\"$ units from $\"P%28-1%2C2%29\"$ , is
\n" ); document.write( " $\"highlight%282x-3y%2B8+%2B-+3sqrt%2839%29=0%29\"$
\n" ); document.write( "b) Line $\"2x-3y=6\"$ <--> $\"2x-6=3y\"$ <--> $\"y=%282%2F3%29x-6%2F3\"$ <--> $\"y=%282%2F3%29x-2\"$
\n" ); document.write( "has a slope of $\"2%2F3\"$ .
\n" ); document.write( "Lines perpendicular to $\"2x-3y=6\"$ have a slope of
\n" ); document.write( " $\"%28-1%29%2F%282%2F3%29=%28-1%29%2A%283%2F2%29=-3%2F2\"$ .
\n" ); document.write( "Their equation will be
\n" ); document.write( " $\"y=-%283%2F2%29x%2Bb\"$ <--> $\"2y=-3x%2B2b\"$ <--> $\"3x%2B2y-2b=0\"$ ,
\n" ); document.write( "which we could write as $\"3x%2B2y%2Bc=0\"$ .
\n" ); document.write( "As before
\n" ); document.write( "

\n" ); document.write( " $\"d%283x%2B2y%2Bc=0%2CP%28-1%2C2%29%29=abs%28-3%2B4%2Bc%29%2Fsqrt%289%2B4%29\"$
\n" ); document.write( " $\"d%283x%2B2y%2Bc=0%2CP%28-1%2C2%29%29=abs%28c%2B1%29%2Fsqrt%2813%29\"$ ,
\n" ); document.write( "and since that distance must be $\"2sqrt%283%29\"$ ,
\n" ); document.write( " $\"abs%28c%2B1%29%2Fsqrt%2813%29=2sqrt%283%29\"$ --> $\"abs%28c%2B1%29=2sqrt%283%29%2Asqrt%2813%29\"$ --> $\"abs%28c%2B1%29=2sqrt%2839%29\"$ --> $\"c=-1+%2B-+2sqrt%2839%29\"$ .
\n" ); document.write( "So the equation of the two lines perpendicular to $\"2x-3y=6\"$ ,
\n" ); document.write( "and passing at a distance $\"2sqrt%283%29\"$ units from $\"P%28-1%2C2%29\"$ , is
\n" ); document.write( " $\"highlight%283x%2B2y-1+%2B-+3sqrt%2839%29=0%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "THE PICTURES AND EXPLANATION FOR THE SITUATION:
\n" ); document.write( "The line represented by $\"2x-3y=6\"$ and
\n" ); document.write( "the circle that is the locus of all the points $\"2sqrt%283%29\"$ units from $\"P%28-1%2C2%29\"$
\n" ); document.write( "are shown below.
\n" ); document.write( "

The circle is centered at $\"p\"$ and has a radius of $\"2sqrt%283%29\"$ units.The equation for the circle is $\"%28x%2B1%29%5E2%2B%28y-2%29%5E2=%282sqrt%283%29%29%5E2\"$ <--> $\"%28x%2B1%29%5E2%2B%28y-2%29%5E2=12\"$ .
\n" ); document.write( "There is more than one line
\n" ); document.write( "parallel to $\"2x-3y=6\"$ and passing at a distance $\"2sqrt%283%29\"$ units from $\"P%28-1%2C2%29\"$ ,
\n" ); document.write( "and there is more than one line
\n" ); document.write( "perpendicular to 2x-3y=6 and passing at a distance $\"2sqrt%283%29\"$ units from $\"P%28-1%2C2%29\"$ .
\n" ); document.write( "The way I read/interpret the problem, we want lines a) parallel, and b) perpendicular to $\"2x-3y=6\"$ , and tangent to the circle $\"%28x%2B1%29%5E2%2B%28y-2%29%5E2=12\"$ .
\n" ); document.write( "Only $\"1\"$ point of each of those lines is at a distance $\"2sqrt%283%29\"$ units from $\"P%28-1%2C2%29\"$ .
\n" ); document.write( "That point is the intersection of the line with the circle $\"%28x%2B1%29%5E2%2B%28y-2%29%5E2=12\"$ .
\n" ); document.write( "The remaining point on each of the lines is at a distance greater than $\"2sqrt%283%29\"$ units from $\"P%28-1%2C2%29\"$ ,
\n" ); document.write( "and therefore is outside the circle of radius.
\n" ); document.write( "
\n" ); document.write( "There are two lines parallel to $\"2x-3y=6\"$ ,
\n" ); document.write( "and tangent to the circle $\"%28x%2B1%29%5E2%2B%28y-2%29%5E2=12\"$ .
\n" ); document.write( "They are tangent at points $\"Q\"$ and $\"R\"$ .
\n" ); document.write( "There are two lines perpendicular to $\"2x-3y=6\"$ ,
\n" ); document.write( "and tangent to the circle $\"%28x%2B1%29%5E2%2B%28y-2%29%5E2=12\"$ .
\n" ); document.write( "They are tangent at points $\"S\"$ and $\"T\"$ .
\n" ); document.write( "

Each of those lines is perpendicular to the radius at the point of tangency.
\n" ); document.write( "So PQ and PR are perpendicular to the two lines parallel to $\"2x-3y=6\"$ and must be perpendicular to $\"2x-3y=6\"$ itself.
\n" ); document.write( "and PS and PT are perpendicular to the two lines perpendicular to $\"2x-3y=6\"$ and must be parallel to $\"2x-3y=6\"$ . \n" ); document.write( "

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