Find the equation of the asymptotes of the hyperbola `3x^2+10 x y+9y^2+14 x+22 y+7=0`
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Since the equation of hyperbola and the combined equation of its asymptotes differ by a constant, the equations of the asymptotes should be `3x^(2)+10xy+8y^(2)+14x+22y+lambda=0" (1)"` Now, `lambda` is to be chosen so that (1) represents a pair of straight lines. Comparing (1) with `ax^(2)+2hxy+by^(2)+2gx+2fy+c=0" (2)"` we have `a=3,b=8,h=5,g=7,f=11, c=lambda` We know that (2) represents a pair of straight lines if `abc+2hgf-af^(2)-bg^(2)-ch^(2)=0` `"or "3xx8xxlambda+2xx7xx11xx5-3xx121-8xx49-lambdaxx25=0` `"or "lambda=15` Hence, the combined equation of the asymptotes is `3x^(2)+10xy+8y^(2)+14x+22y+15=0`.
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