Consider the family of lines $a(3x+4y+6)+b(x+y+2)=0$ Find the equation…











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Question :



Consider the family of lines $a(3x+4y+6)+b(x+y+2)=0$ Find the equation of the line of family situated at the greatest distance from the point P (2,3)



Solution :



The given equation can be written as $(3x+4y+6)+lambda (x+y+2)=0$
$Rightarrow x(3+lambda)+y(4+lambda)+6+2lambda =0....(1)$



Distance of point P(2,3) from the above line (1) is given by



D= $frac{|2(3+lambda)+3(4+lambda)+6+2lambda|}{sqrt{(3+lambda)^2+(4+lambda)^2}}$



$Rightarrow D = frac{(24+7lambda)^2}{(3+lambda)^2+(4+lambda)^2}$



Now how to maximize the aboved distance please suggest. Thanks






geometry optimization coordinate-systems







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    up vote
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    down vote

    favorite
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    Question :



    Consider the family of lines $a(3x+4y+6)+b(x+y+2)=0$ Find the equation of the line of family situated at the greatest distance from the point P (2,3)



    Solution :



    The given equation can be written as $(3x+4y+6)+lambda (x+y+2)=0$
    $Rightarrow x(3+lambda)+y(4+lambda)+6+2lambda =0....(1)$



    Distance of point P(2,3) from the above line (1) is given by



    D= $frac{|2(3+lambda)+3(4+lambda)+6+2lambda|}{sqrt{(3+lambda)^2+(4+lambda)^2}}$



    $Rightarrow D = frac{(24+7lambda)^2}{(3+lambda)^2+(4+lambda)^2}$



    Now how to maximize the aboved distance please suggest. Thanks






    geometry optimization coordinate-systems







    share|cite|improve this question


























      up vote
      6
      down vote

      favorite
      3









      up vote
      6
      down vote

      favorite
      3






      3





      Question :



      Consider the family of lines $a(3x+4y+6)+b(x+y+2)=0$ Find the equation of the line of family situated at the greatest distance from the point P (2,3)



      Solution :



      The given equation can be written as $(3x+4y+6)+lambda (x+y+2)=0$
      $Rightarrow x(3+lambda)+y(4+lambda)+6+2lambda =0....(1)$



      Distance of point P(2,3) from the above line (1) is given by



      D= $frac{|2(3+lambda)+3(4+lambda)+6+2lambda|}{sqrt{(3+lambda)^2+(4+lambda)^2}}$



      $Rightarrow D = frac{(24+7lambda)^2}{(3+lambda)^2+(4+lambda)^2}$



      Now how to maximize the aboved distance please suggest. Thanks






      geometry optimization coordinate-systems







      share|cite|improve this question















      Question :



      Consider the family of lines $a(3x+4y+6)+b(x+y+2)=0$ Find the equation of the line of family situated at the greatest distance from the point P (2,3)



      Solution :



      The given equation can be written as $(3x+4y+6)+lambda (x+y+2)=0$
      $Rightarrow x(3+lambda)+y(4+lambda)+6+2lambda =0....(1)$



      Distance of point P(2,3) from the above line (1) is given by



      D= $frac{|2(3+lambda)+3(4+lambda)+6+2lambda|}{sqrt{(3+lambda)^2+(4+lambda)^2}}$



      $Rightarrow D = frac{(24+7lambda)^2}{(3+lambda)^2+(4+lambda)^2}$



      Now how to maximize the aboved distance please suggest. Thanks






      geometry optimization coordinate-systems




      geometry optimization coordinate-systems






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      share|cite|improve this question













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      edited Dec 14 '15 at 19:07









      G-man

      4,47331344




      4,47331344










      asked Oct 16 '14 at 8:03









      user108258

      542819




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          First, note that all the lines that can be represented by your equation pass through the point (-2,0). Obviously it follows that the foot of the perpendicular to the line from (2,3) should be (-2,0)






          share|cite|improve this answer























          • I don't see it as "obvious" to be honest... Take $a=0$.
            – Martigan
            Oct 16 '14 at 9:06




















          up vote
          1
          down vote













          Find the common point of intersection of this family of lines. Here it is (-2,0). Now, given point is (2,3). Equation of line passing through both the points is : 3x + 6 =4y. You actually only need the slope of this line which is : 3/4 Line perpendicular to this line passing through the common point will be at the greatest distance. Slope of the required line: -4/3Equation of the line :-4/3{x-(-2)} = y-0After solving, equation is : 4x + 3y + 8 =0.






          share|cite|improve this answer





















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            2 Answers
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            active

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            2 Answers
            2






            active

            oldest

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            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            3
            down vote













            First, note that all the lines that can be represented by your equation pass through the point (-2,0). Obviously it follows that the foot of the perpendicular to the line from (2,3) should be (-2,0)






            share|cite|improve this answer























            • I don't see it as "obvious" to be honest... Take $a=0$.
              – Martigan
              Oct 16 '14 at 9:06

















            up vote
            3
            down vote













            First, note that all the lines that can be represented by your equation pass through the point (-2,0). Obviously it follows that the foot of the perpendicular to the line from (2,3) should be (-2,0)






            share|cite|improve this answer























            • I don't see it as "obvious" to be honest... Take $a=0$.
              – Martigan
              Oct 16 '14 at 9:06















            up vote
            3
            down vote










            up vote
            3
            down vote









            First, note that all the lines that can be represented by your equation pass through the point (-2,0). Obviously it follows that the foot of the perpendicular to the line from (2,3) should be (-2,0)






            share|cite|improve this answer














            First, note that all the lines that can be represented by your equation pass through the point (-2,0). Obviously it follows that the foot of the perpendicular to the line from (2,3) should be (-2,0)







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Oct 25 '14 at 12:31

























            answered Oct 16 '14 at 9:01









            G-man

            4,47331344




            4,47331344












            • I don't see it as "obvious" to be honest... Take $a=0$.
              – Martigan
              Oct 16 '14 at 9:06




















            • I don't see it as "obvious" to be honest... Take $a=0$.
              – Martigan
              Oct 16 '14 at 9:06


















            I don't see it as "obvious" to be honest... Take $a=0$.
            – Martigan
            Oct 16 '14 at 9:06






            I don't see it as "obvious" to be honest... Take $a=0$.
            – Martigan
            Oct 16 '14 at 9:06












            up vote
            1
            down vote













            Find the common point of intersection of this family of lines. Here it is (-2,0). Now, given point is (2,3). Equation of line passing through both the points is : 3x + 6 =4y. You actually only need the slope of this line which is : 3/4 Line perpendicular to this line passing through the common point will be at the greatest distance. Slope of the required line: -4/3Equation of the line :-4/3{x-(-2)} = y-0After solving, equation is : 4x + 3y + 8 =0.






            share|cite|improve this answer

























              up vote
              1
              down vote













              Find the common point of intersection of this family of lines. Here it is (-2,0). Now, given point is (2,3). Equation of line passing through both the points is : 3x + 6 =4y. You actually only need the slope of this line which is : 3/4 Line perpendicular to this line passing through the common point will be at the greatest distance. Slope of the required line: -4/3Equation of the line :-4/3{x-(-2)} = y-0After solving, equation is : 4x + 3y + 8 =0.






              share|cite|improve this answer























                up vote
                1
                down vote










                up vote
                1
                down vote









                Find the common point of intersection of this family of lines. Here it is (-2,0). Now, given point is (2,3). Equation of line passing through both the points is : 3x + 6 =4y. You actually only need the slope of this line which is : 3/4 Line perpendicular to this line passing through the common point will be at the greatest distance. Slope of the required line: -4/3Equation of the line :-4/3{x-(-2)} = y-0After solving, equation is : 4x + 3y + 8 =0.






                share|cite|improve this answer












                Find the common point of intersection of this family of lines. Here it is (-2,0). Now, given point is (2,3). Equation of line passing through both the points is : 3x + 6 =4y. You actually only need the slope of this line which is : 3/4 Line perpendicular to this line passing through the common point will be at the greatest distance. Slope of the required line: -4/3Equation of the line :-4/3{x-(-2)} = y-0After solving, equation is : 4x + 3y + 8 =0.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 22 at 3:12









                Heet

                111




                111






























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