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
edited Dec 14 '15 at 19:07
G-man
4,47331344
asked Oct 16 '14 at 8:03
user108258
542819
add a comment |
up vote
6
down vote
favorite
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
edited Dec 14 '15 at 19:07
G-man
4,47331344
asked Oct 16 '14 at 8:03
user108258
542819
add a comment |
up vote
6
down vote
favorite
up vote
6
down vote
favorite
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
edited Dec 14 '15 at 19:07
G-man
4,47331344
asked Oct 16 '14 at 8:03
user108258
542819
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
edited Dec 14 '15 at 19:07
G-man
4,47331344
asked Oct 16 '14 at 8:03
user108258
542819
edited Dec 14 '15 at 19:07
G-man
4,47331344
asked Oct 16 '14 at 8:03
user108258
542819
edited Dec 14 '15 at 19:07
G-man
4,47331344
edited Dec 14 '15 at 19:07
G-man
4,47331344
edited Dec 14 '15 at 19:07
G-man
4,47331344
4,47331344
asked Oct 16 '14 at 8:03
user108258
542819
asked Oct 16 '14 at 8:03
user108258
542819
asked Oct 16 '14 at 8:03
user108258
542819
542819
add a comment |
add a comment |
2 Answers
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3
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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)
answered Oct 16 '14 at 9:01
G-man
4,47331344
I don't see it as "obvious" to be honest... Take $a=0$.
– Martigan
Oct 16 '14 at 9:06
add a comment |
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.
answered Nov 22 at 3:12
Heet
111
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
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)
answered Oct 16 '14 at 9:01
G-man
4,47331344
I don't see it as "obvious" to be honest... Take $a=0$.
– Martigan
Oct 16 '14 at 9:06
add a comment |
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)
answered Oct 16 '14 at 9:01
G-man
4,47331344
I don't see it as "obvious" to be honest... Take $a=0$.
– Martigan
Oct 16 '14 at 9:06
add a comment |
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)
answered Oct 16 '14 at 9:01
G-man
4,47331344
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)
answered Oct 16 '14 at 9:01
G-man
4,47331344
edited Oct 25 '14 at 12:31
answered Oct 16 '14 at 9:01
G-man
4,47331344
answered Oct 16 '14 at 9:01
G-man
4,47331344
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
add a comment |
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
add a comment |
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.
answered Nov 22 at 3:12
Heet
111
add a comment |
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.
answered Nov 22 at 3:12
Heet
111
add a comment |
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.
answered Nov 22 at 3:12
Heet
111
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.
answered Nov 22 at 3:12
Heet
111
answered Nov 22 at 3:12
Heet
111
answered Nov 22 at 3:12
Heet
111
answered Nov 22 at 3:12
Heet
111
111
add a comment |
add a comment |
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