What is the proof of the formula L1 +pL2=0?

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Homework Statement

This related to coordinate geometry.
According to my book it says if there are two intersecting lines L₁(a₁x+b₁y+c₁=0) and L₂(a₂x+b₂y+c₂=0)
Then equation of any line passing through their point of intersection is
L₁ +p L₂ =0
or (a₁x+b₁y+c₁=0) +p(a₂x+b₂y+c₂=0)=0
where p is some constant
my problem is that i don't understand what is the proof of this formula.Book also says that (a₁x+b₁y+c₁=0) +p(a₂x+b₂y+c₂=0)=0 represents a Family of lines , and i don't understand what it means too.

Homework Equations

The Attempt at a Solution

i tried to use some arbitrary line equations and tried to simultaneously solve the equations
and get its point of intersection and find the line's equation but i can't relate these 2 methods
If k is such that it a₁-p.a₂=0 how does this proove these lines are concurrent and it equation is (a₁x+b₁y+c₁=0) +p(a₂x+b₂y+c₂=0)=0
book gives the proof as since this line satisfies some P(a,b) hence when we put it into the equation it will yield 0
But that comes only after i have proved this is the equation of all lines passing through common point ,how can this be a proof?

Could you please give me a proof of how this formula is derived?
Thank you.

 

[hamsterman]

You have a line Lp defined by, (a1x+b1y+c1) +p(a2x+b2y+c2)=0. Say that lines L1, L2 intersect at a point P, then P is both on L1 and on L2, thus a1*Px+b1*Py+c1 = 0 and a2*Px+b2*Py+c2 = 0. 0+p*0 = 0 so P is also on Lp. That's all there is to it.

A family of lines means that Lp is not any specific line. It could represent many different lines, if different values of p are used.

 

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You have a line Lp defined by, (a1x+b1y+c1) +p(a2x+b2y+c2)=0. Say that lines L1, L2 intersect at a point P, then P is both on L1 and on L2, thus a1*Px+b1*Py+c1 = 0 and a2*Px+b2*Py+c2 = 0. 0+p*0 = 0 so P is also on Lp. That's all there is to it.

I already know if P satisfies L1 and L2 it is on line L1 and L2 because basically equation of line L1 and L2 is a condition & when we graph it we get a line.But what i am asking is HOW DO WE KNOW EQUATION OF LINES PASSING THROUGH A POINT CAN BE GIVEN BY (a₁x+b₁y+c₁=0) +p(a₂x+b₂y+c₂=0)=0
I WANT THE PROOF .

One more thing i found was if we add or subtract the equations of two intersecting lines the resultant line will be concurrent to the other two lines
And if it is correct(which i hope it is(!)) then how can find if they are concurrent from their equations when the slope and the y-intercept always remains the same no matter which point on line we take.

 

[hamsterman]

In the line equation ax + by + c = 0, (a, b) is a normal of the line. Another way to write the same equation is <(a, b), (x, y)> = -c, where < , > is the scalar product. It follows from properties of < , > that if <p, q> = w then if you take any vector n, orthogonal to p, <p, q+n> is also = w.

Or was that not what you were asking again?
What I gave you is a valid proof that if L1 and L2 both contain P, then so does L1+p*L2. It is not necessary to derive a thing to prove it.

 

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I don't understand!

 

[hamsterman]

You'll have to be more specific. Do you know what a normal is? A scalar product? That <a, b> = 0 if a and b are perpendicular? That <a+b, c> = (ax+bx)*cx + (ay+by)*cy = ax*cx + ay*cy + bx*cx + by*cy = <a, c> + <b, c>?