A locus question

Question

A is a give point and P is any point on a given straight line. If AQ=AP and AQ makes a constant angle with AP find the locus of Q.

The answer should be a line ,but i do not know how to prove it .

Can someone pls help

Answer 1

So, essentially you are saying that $Q$ is obtained by rotating $P$ around $A$ a fixed angle $\alpha$. Then, the locus of $Q$ is the line that you obtain after rotating your given line an angle $\alpha$ around $A$.

Edit: If we want to be more formal, let's make a coordinate system centered at $A$, i.e. consider $A$ to be the origin $(0,0)$. Let's say that the given line has equation $aX+bY=c$. Now, given a point $P=(x,y)$ in this line (so, $ax+by=c$) we have that $Q$ is the point obtained after rotating $P$ around $A$ an angle of $\alpha$, this is given by $Q=(x',y')=(x\cos\alpha-y\sin\alpha,x\sin\alpha+y\cos\alpha )$. Since $x=x'\cos\alpha+y'\sin\alpha$ and $y=y'\cos\alpha-x'\sin\alpha$ we conclude that $$a(x'\cos\alpha+y'\sin\alpha)+b(y'\cos\alpha-x'\sin\alpha)=c$$ Therefore, $(a\cos\alpha-b\sin\alpha)x'+(a\sin\alpha+b\cos\alpha)y'=c$. Which means that the locus of $Q$ is the line with equation $(a\cos\alpha-b\sin\alpha)X+(a\sin\alpha+b\cos\alpha)Y=c$.

Appendix: Here I show how we get the coordinates $Q=(x',y')$ that appear after rotating $P=(x,y)$ an angle of $\alpha$. Let's say $(x,y)$ has polar coordinates $(r,\theta)$ (i.e. $r$ is the distance from $(x,y)$ to the origin $A$ and $\theta$ is the angle between the $X$-axis and $AP$). We have $$(x,y)=(r\cos\theta,r\sin\theta)$$ Now, $Q=(x',y')$ will have polar coordinates $(r,\theta+\alpha)$, so $$(x',y')=(r\cos(\theta+\alpha),r\sin(\theta+\alpha))$$ Using $\cos(\theta+\alpha)=\cos\theta\cos\alpha-\sin\theta\sin\alpha$, you get $$x'=r\cos(\theta+\alpha)=r\cos\theta\cos\alpha-r\sin\theta\sin\alpha=x\cos\alpha-y\sin\alpha$$ Similarly, using that $\sin(\theta+\alpha)=\sin\theta\cos\alpha+\cos\theta\sin\alpha$, we get $$y'=r\sin(\theta+\alpha)=r\sin\theta\cos\alpha+r\cos\theta\sin\alpha=y\cos\alpha+x\sin\alpha$$