Question
Two lines represented by the equation $${x^2} - {y^2} - 2x + 1 = 0$$ are rotated about the point $$\left( {1,\,0} \right),$$ the line making the bigger angle with the positive direction of the $$x$$-axis being turned by $${45^ \circ }$$ in the clockwise sense and the other line being turned by $${15^ \circ }$$ in the anticlockwise sense. The combined equation of the pair of lines in their new positions is :
A.
$$\sqrt 3 {x^2} - xy + 2\sqrt 3 x - y + \sqrt 3 = 0$$
B.
$$\sqrt 3 {x^2} - xy - 2\sqrt 3 x + y + \sqrt 3 = 0$$
C.
$$\sqrt 3 {x^2} - xy - 2\sqrt 3 x + \sqrt 3 = 0$$
D.
none of these
Answer :
$$\sqrt 3 {x^2} - xy - 2\sqrt 3 x + y + \sqrt 3 = 0$$
Solution :
The combined equation is $$\left( {x + y - 1} \right)\left( {x - y - 1} \right) = 0.$$ After rotation, the lines are $$x = 1$$ and $$y = \sqrt 3 \left( {x - 1} \right).$$ So, the combined equation after rotation is $$\left( {x - 1} \right)\left( {\sqrt 3 x - y - \sqrt 3 } \right) = 0.$$
The combined equation is $$\left( {x + y - 1} \right)\left( {x - y - 1} \right) = 0.$$ After rotation, the lines are $$x = 1$$ and $$y = \sqrt 3 \left( {x - 1} \right).$$ So, the combined equation after rotation is $$\left( {x - 1} \right)\left( {\sqrt 3 x - y - \sqrt 3 } \right) = 0.$$