Find the equation to the pair of angle bisectors of the pair of lines ( a x + b y ) 2 = 3 ( b x − a y ) 2 .Efforts: ( a x + b y ) 2 = 3 ( b x − a y ) 2 After simplifying, I got: x 2 ( a 2 − 3 b 2 ) + 8 a b x y + y 2 ( b 2 − 3 a 2 ) = 0Now, what should I do next?

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Find the equation to the pair of angle bisectors of the pair of lines   (  a  x  +  b  y      )    2    =  3  (  b  x  −  a  y      )    2  .Efforts:  (  a  x  +  b  y      )    2    =  3  (  b  x  −  a  y      )    2  After simplifying, I got:      x    2    (      a    2    −  3      b    2    )  +  8  a  b  x  y  +      y    2    (      b    2    −  3      a    2    )  =  0Now, what should I do next?

Cahokiavv · Accepted Answer

Taking the square root on both sides, we obtain  a  x  +  b  y  =  −      3    (  b  x  −  a  y  )    and    a  x  +  b  y  =      3    (  b  x  −  a  y  )and the two lines are  (  a  +      3    b  )  x  +  (  b  −      3    a  )  y  =  0    and    (  a  −      3    b  )  x  +  (  b  +      3    a  )  y  =  0.The equation for the bisectors is known to be                    |            (      a      +              3            b      )      x      +      (      b      −              3            a      )      y              |                    (      a      +              3            b              )        2            +      (      b      −              3            a              )        2              =                    |            (      a      −              3            b      )      x      +      (      b      +              3            a      )      y              |                    (      a      −              3            b              )        2            +      (      b      +              3            a              )        2              ,which in fact is equivalent to      |    (  a  +      3    b  )  x  +  (  b  −      3    a  )  y      |    =      |    (  a  −      3    b  )  x  +  (  b  +      3    a  )  y      |    .This gives  b  x  −  a  y  =  0    or    a  x  +  b  y  =  0.

Gaaljh · Accepted Answer

You can simplify and try the hard way. Here&#039;s what I will do: Here:  (  a  x  +  b  y      )    2    −  3  ∗  (  b  x  −  a  y      )    2    =  0this is of the form      a    2    −      b    2    =  (  a  +  b  )  (  a  −  b  )=&amp;gt; the two lines are  (  a  x  +  b  y  +  s  q  r  t  (  3  )  ∗  b  x  −  s  q  r  t  (  3  )  ∗  a  y  =  0  )and  (  a  x  +  b  y  −  s  q  r  t  (  3  )  ∗  b  x  +  s  q  r  t  (  3  )  ∗  a  y  =  0  )

Find the equation to the pair of angle bisectors of the pair of lines ( a x + b y

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