Question

Family of lines represented by the equation $(\cos \theta+\sin$ $\theta) x +(\cos \theta-\sin \theta) y -3(3 \cos \theta+\sin \theta)=0$ passes through a fixed point $M$ for all real values of $\theta$. The reflection of $M$ in the line $x-y=0$, is

• A. (6, 3)
• B. (3, 6)
• C. (-6, 3)
• D. (3, -6)

Answer 1

Answer: (B)

Given equation is $\cos \theta(x+y-9)+\sin \theta(x-y-3)=0$
$\Rightarrow (x+y-9)+\tan \theta(x-y-3)=0$, which is of the form $L _{1}+\lambda L _{2}=0$
where $L_{1}: x+y-9=0$ and $L_{2}: x-y-3=0$
Hence the lines will always pass through the point of intersection of $L _{1}=0$ and $L _{2}=0$ i.e., $M (6,3)$
So its reflection in the line $y = x$ will be $(3,6)$