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If the pairs of straight lines x^{2}-2pxy-y^{2}=0\; and\; x^{2}-2qxy-y^{2}=0\;  be such that each pair bisects the angle between the other pair, then

  • Option 1)


  • Option 2)

    \; pq=1\;

  • Option 3)

    \; pq=-1\;

  • Option 4)

    \; p=q


Answers (1)


As we learnt in

General equation of a conic -

ax^{2}+2hxy+by^{2}+2gx+2fy+c= 0

- wherein

a,b,c, f,g,h  are constants



and x^{2}-2pxy-y^{2}=0

They bisect angle between the other pair

Equation of bisector is given by

\frac{x^{2}-y^{2}}{1-(-1)}=\frac{xy}{-p} \Rightarrow x^{2}+\frac{2}{p}xy -y^{2}=0

So \frac{2}{p}=-2q\Rightarrow pq =-1


Option 1)


This option is incorrect.

Option 2)

\; pq=1\;

This option is incorrect.

Option 3)

\; pq=-1\;

This option is correct.

Option 4)

\; p=q

This option is incorrect.

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