If $$A(1,3)$$ and $$C(7,5)$$ are two opposite vertices of a square, then find the equation of a side passing through $$A$$.

$$C$$ is the centroid of the triangle with vertices $$(3,-1),(1,3)$$ and $$(2,4)$$. Let $$P$$ be the point of intersection of the lines $$x+3 y-1=0$$ and $$3 x-y+1=0$$. Then a line which passes through both points $$C$$ and $$P$$ would also passes through the point .......

The distance of the point $$(1,2)$$ from the line $$x+y+5=0$$ measured along the line parallel to $$3 x-y=7$$ is equal to

Find the equation of a line which passes through $$\left(2 \cos ^3(\theta), 2 \sin ^3(\theta)\right)$$ and is perpendicular to the line $$x \cos (\theta)-y \sin (\theta)=2 \cos (2 \theta)$$.