What Is the Use of Families of Straight Lines in Geometry?

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Families of straight lines are used in geometry to represent lines passing through the intersection of two given lines, L1 and L2. The equation L1 + kL2 = 0, where k is a variable, describes a family of lines that includes L1 but not L2 itself. This is because there is no value of k that can make L2 equal to L1 + kL2. To represent all lines through the intersection point of L1 and L2, the equation l(A1x1 + B1y1 + C1) + m(A2x1 + B2y1 + C2) = 0 is used, where l and m are real numbers. This approach ensures that all lines through the intersection are included in the family.
theow
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This is my second post on PF =)
I want to ask what is the use of Families of straight lines?
I am thinking of A Family of Straight Lines Passing Through the Intersection of Two Lines.
We have the equation: L1+kL2=0 where L1=L2=0 and k is a variable, right?
But is it said that L2 is not included in this family?
So why are we using this equation, when it cannot fully represent all the lines with this common properties, so as to classify them into a family?
Please, may you help. Thanks
 
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"We have the equation: L1+kL2=0 where L1=L2=0 and k is a variable, right?"

I cannot make sense out of that sentence. If "L1= L2= 0 and k is a variable" then the equation L1+ kL2= 0 just says "0= 0". It says nothing about any "family of straight lines".
 
HallsofIvy said:
"We have the equation: L1+kL2=0 where L1=L2=0 and k is a variable, right?"

I cannot make sense out of that sentence. If "L1= L2= 0 and k is a variable" then the equation L1+ kL2= 0 just says "0= 0". It says nothing about any "family of straight lines".

Maybe I wasn't making the question clear enough...

Here's what I find in my textbook:

Given two straight lines
L1: A1x+B1y+C1=0
and L2: A2x+B2y+C2=0
which intersects at the point P(x1,y1)
Substitute P(x1,y1) into L1 and L2 respectively, we have
A1x1+B1y1+C1=0...(1)
A2x1+B2y1+C2=0...(2)
Consider
L: (A1x1+B1y1+C1)+k(A2x1+B2y1+C2)=0, where k is real.
For each value of k, together with (1) and (2), we have
(A1x1+B1y1+C1)+k(A2x1+B2y1+C2)=0+k(0)=0
which shows that L passes through P.
L can also be arranged as
(A1+kA2)x+(B1+kB2)y+(C1+kC2)=0
which shows that L is a straight line.
In conclusion, as k varies,
(A1x1+B1y1+C1)+k(A2x1+B2y1+C2)=0, where k is real,
represent a family of straight lines passing through the point of intersection of L1 and L2.
It should be emphasized that the line L2 is not included in this family. In order to represent all the lines passing through the point of intersection of L1 and L2, th efolloewing form would be used:
l(A1x1+B1y1+C1)+m(A2x1+B2y1+C2)=0, where l and m are real.

So why don't we use the last equation instead?
Thanks.
 
Welcome to PF!

Hi theow ! Welcome to PF! :smile:
theow said:
L1 + kL2

But is it said that L2 is not included in this family?

Yes … L1 is included, because L1 = L1 + kL2 with k= 0.

But there is no k (unless you include infinity, which is not allowed) for which L2 = L1 + kL2, is there? :smile:
theow said:
So why don't we use the last equation instead?

We do … your textbook says:
In order to represent all the lines passing through the point of intersection of L1 and L2, the following form would be used:
l(A1x1+B1y1+C1)+m(A2x1+B2y1+C2)=0, where l and m are real.

We use lL1 + mL2. :smile:
 
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