I got two equations of a line a1x + b1y + c1 =0 and a2x + b2y + c2 =0 they multiplied both and said gave a quadratic staying it represents a system of lines. it's a big quadratic(in x and y) with a lot of coefficients and right below it they said it intersects at some blah point (a, b) where a and b are functions of the coefficients of the quadratic. I tried to prove this by saying the point of intersection will give you the maximum value of y and x in the quadratic. to do this i total diffeitated (not partial) wrt x and then wrt y and set the dy/dx and dx/dy to 0, then I and got 2 linear equations i solved it and got the right answer (a, b) .
I have no idea what i just did : |
The Attempt at a Solution
to convince myself that the maximum value of y this really is the case i took y= m1x + c1 and y = m2x+c2 corresponding to the above a1x+b1y + c1 and a2x + b2y+c2 and multiplied it getting y^2 = m1m2x^2 + x(m1c2+m2c1) + c1c2. differtiation with respect to y and setting it do 0 does not give me the point of intersection : (
what on Earth is going on?!
*Usually I see partial differtial equations for maximizing 2 variables (no idea how that works either) but this is total? why is it working? what's even happening?
* If the first form gives me the right answer, why not with y = mx +c?
*general case wot.jpg
pic attached is the problematic nonsense. second pic is two equations arrived after differtiation and solving by crammer's to get the right answer (in the corner)