Question

Two lines represented by the equation $x^2-y^2-2x+1=0$ are rotated about the point $(1,0)$, 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 ${150}^{\circ }$ 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 1

The correct option is B $\sqrt{3}{x}^2-xy-2\sqrt{3}x+y+\sqrt{3}=0$

$x^2-y^2-2x+1=0$

$\Rightarrow (x^2-2x+1)-y^2=0$

$\Rightarrow {(x-1)}^2-y^2=0$

$\Rightarrow (x-y-1)(x+y-1)=0$

which is the combined equation

After rotation the lines become $x=1$ and $y=\sqrt{3}(x-1)$

So the combined equation after rotation is

$(x-1)(\sqrt{3}x-y-\sqrt{3})=0$

$\Rightarrow \sqrt{3}{x}^2-xy-2\sqrt{3}x+\sqrt{3}+y=0$