show that two straight lines through the origin which make angles of 45 degree with straight line lx+my+n=0 are given by (l^2-m^2)(x^2-y^2)+4lmxy=0

Question

ganeshie8 · Answer

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ganeshie8 · Answer

Notice that the lines thru origin must intersect at 90 degrees.

say the lines are :

$\large y = kx$

$\large y = \frac{-1}{k}x$

ganeshie8 · Answer

combining both gives the pair of lines :

$\large  (y-kx)(y + \frac{1}{k}x) =  0$

ganeshie8 · Answer

you can find the value of  $k$   by using  the fact that these lines make 45 degrees angle wid $lx + my + n = 0$   :

$\large    1 =   \frac{k - \frac{-l}{m}}{1 + k(\frac{-l}{m})}$


$\large k =   \frac{m-l}{m+l}$

ganeshie8 · Answer

plug this value in the pair of lines equation and simplify

anonymous · Answer

thanks.....