A line through the point $(1,0)$ meets the variable line $y=tx$ at right angle at point $P.$
Find in terms of $t$,the coordinates of $P.$
I’ve found the coordinates of $P$ to be $\displaystyle\Big(\frac{1}{1+t^2},\frac{t}{1+t^2}\Big)$
Show that the locus of $P$ as $t$ varies is a circle and state its centre and radius.
How to show that the locus of $P$ as $t$ varies is a circle?
You have the variable coordinates of $P$ : $$x=\frac{1}{t^2+1}$$ $$y=\frac{t}{t^2+1}$$
As Jan-Magnus said, to prove that the locus is a circle, it’s a good idea to square and add $x$ and $y$.
$$x^2+y^2 = \frac{1+t^2}{(1+t^2)^2} = \frac{1}{1+t^2}$$ $$x^2 + y^2 - x=0$$ or $$\left(x-\frac12\right)^2 + y^2 = \left(\frac 12 \right)^2$$
which tells us that the center of the circle is $\left(\frac 12,0 \right)$ and the radius is $\frac 12$.