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Find the condition that the expressions ...

Find the condition that the expressions `a x^2-b x y+c y^2a n da_1x^2+b_1x y+c_1y^2` may have factors `y-m xa n dm y-x ,` respectively.

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y = mx is a factor of ` ax^(2) + bxy + c^(2)`
Hence, ` ax^(2) + bxy + cy^(2)` will be zero when y - mx = 0 or y = mx
`rArr ax^(2) + bx *mx + cm^(2) x^(2) = 0`
or `cm^(2) + bm + a = 0` (1)
Since my - x is a factor of ` a_(1)x^(2) + b_(1)xy + C_(1) y^(2) = 0` when my - x = 0
`rArr a_(1) m^(2) y^(2) + b_(1)*my* + c_(1)y^(2) = 0` [Putting x = my]
`rArr a_(1) m^(2) + b_(1) m + c_(1) = 0` (2)
Eliminating m from (1) and (2) , we get
`(bc_(1) - ab_(1)) (cb_(1) - ba_(1)) = (aa_(1) - c c_(1))^(2)` .
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