Here is the question. Find two lines L1, L2 that intersect at point (a,b). So how do yo go about it? Any help is appreciated, thanks.
But you need to find out at least the slope of the line to get it, don't you? And since you don't have the y intercept of the line, that would also be difficult to get, isn't it? So you have two lines intersecting at point (6,18) and you don't have the gradient (slope) or the y intercept. You are just given the point of intersection of x and y where x = 6 and y = 18 as I said above.
Thanks but I just want to understand more. 2 and 3 are the slopes for L1 and L2 respectively? If so, can you please tell me how you got because I want to know. So the equation will be y =2x-2a+b for L1 and y=3x-3a+b for L2? And what are a and b? Lastly, s it it not possible for us to get the equation in the form y=mx+b? Thank you and sorry for asking too many questions.
Ok but the equatio says that y intrecept is b minus the product of the slope and a. My question is, what do these two variables (a and b) stand for. I mean we know what x, y and m are but what are a and b? Lastly is it not possible for me to give the lines in the usual form of y=mx+b without b being equal to (b-ma)?
Oh OK I get it now. I see. I know the first time I gave the coordinates as (a,b) but later on I put them as (6,18). So let us say we have the intersection point as (6,18). Using the equation which you gave which was:
$\displaystyle L_1:~y-b=2(x-a)$
$\displaystyle L_2:~y-b=3(x-a)$ we get:
L1: y=2x+6
L2: y=3x
Am I right?
Do you understand that there are an infinite number of lines passing through the given point? You only asked for two of them. Any line passing through point $\displaystyle (x_0, y_0)$ can be written in the form $\displaystyle y= m(x- x_0)+ y_0$. Pick any two numbers for m and you get two different lines through $\displaystyle (x_0,y_0)$.