In the picture attached , you can see that the two equations have been made "Homogeneous" . 1. When can we say that two equations are homogeneous? 2. How can we make two equations homogeneous?3. Why is that 2(gy+fx) is multiplied by the term once while c is multiplied twice?
  a x 2 + 2 h x y + b y 2 + 2 g x + 2 f y   c = 0 . . . . . . . i   a n d   s t r a i g h t   l i n e   b e   l x + m y + n = 0 . . . . . . . . . . . . . . i i   N o w   j o i n t   e q u a t i o n   o f   l i n e   O P   a n d   O Q   j o i n i n g   t h e   o r i g i n   a n d   p o i n t s   o f     i n t e r s e c t i o n   P   a n d   Q   c a n   b e   o b t a i n e d   b y   m a k i n g   t h e   e q u a t i o n   ( i )   h o m o g e n o u s   w i t h   t h e   h e l p   o f   e q u a t i o n   o f   t h e   l i n e .   T h u s   r e q u i r e c l   e q u a t i o n   i s   g i v e n   b y               a x 2 + 2 h x y + b y 2 + 2 g x + f y l x + m y - n + c l x + m y - n 2 = 0

Dear Student,

A homogeneous equation is an equation in which  the power of every variable is a non negative integer and the sum of the powers of each term in the expression is same.

As in the case of the equation given in the question:

ax2+2hxy+by2+2gx+2fy+c=0.....(i) 
To make the equation homogeneous we need to make the power of each term in the expression same i.e. 2

Thus, to make the equation homogeneous with lx+my+n=0.....(ii)
lx+my+n=0 can be rewritten as,

lx+my=-nor, lx+my-n=1.....(iii)

Hence, to make the equation (i) homogeneous and to make the sum of powers of each term equal to 2,
terms with power as 1 will be multiplied with equation (iii) once and the constant terms which neither has x or y present with them; will be multiplied with the square of equation (iii),

Thus,

ax2+2hxy+by2+2gx+2fy+c=0or, ax2+2hxy+by2+2gx(1)+2fy(1)+c(1)2=0or, ax2+2hxy+by2+2gxlx+my-n+2fylx+my-n+clx+my-n2=0                    Replacing 1=lx+my-n from (iii)or, ax2+2hxy+by2+2gx+fylx+my-n+clx+my-n2=0

Hope this information will clear your doubts about the topic.

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