If the lines represented by x2-2pxy-y2 are rotated about the origin through an angle θ , one in clockwise direction and other in anti-clockwise direction. Then, die equation of bisectors of the angles between the lines in the new position is

Question

If the lines represented by x2-2pxy-y2 are rotated about the origin through an angle θ , one in clockwise direction and other in anti-clockwise direction. Then, die equation of bisectors of the angles between the lines in the new position is (A)  px2+2xy+py2=0  (B)  px2-2xy+py2=0  (C)  px2+2xy-py2=0  (D) None of these

Tardigrade Admin · Accepted Answer

The bisectors of the angles between the lines in the new position are same as the bisectors of the angles between their old positions. Therefore, required equation is

< b r / > \frac{x ^{2} - y ^{2}}{1 - ( - 1 )} = \frac{x y}{- p}

\Rightarrow - p x^{2} + p y^{2} = 2 x y

\Rightarrow p x^{2} + 2 x y - p y^{2} = 0 < b r / >

Q. If the lines represented by $x^{2} - 2 p x y - y^{2}$ are rotated about the origin through an angle $θ$ , one in clockwise direction and other in anti-clockwise direction. Then, die equation of bisectors of the angles between the lines in the new position is

$p x^{2} + 2 x y + p y^{2} = 0$

$p x^{2} - 2 x y + p y^{2} = 0$

$p x^{2} + 2 x y - p y^{2} = 0$

None of these

Q. If the lines represented by x2−2pxy−y2 are rotated about the origin through an angle θ , one in clockwise direction and other in anti-clockwise direction. Then, die equation of bisectors of the angles between the lines in the new position is

px2+2xy+py2=0

px2−2xy+py2=0

px2+2xy−py2=0

None of these

The bisectors of the angles between the lines in the new position are same as the bisectors of the angles between their old positions. Therefore, required equation is <br/>1−(−1)x2−y2​=−pxy​ ⇒−px2+py2=2xy ⇒px2+2xy−py2=0<br/>

Q. If the lines represented by $x^{2} - 2 p x y - y^{2}$ are rotated about the origin through an angle $θ$ , one in clockwise direction and other in anti-clockwise direction. Then, die equation of bisectors of the angles between the lines in the new position is

$p x^{2} + 2 x y + p y^{2} = 0$

$p x^{2} - 2 x y + p y^{2} = 0$

$p x^{2} + 2 x y - p y^{2} = 0$