Questions on the use of algebraic techniques to prove geometric theorems.

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Single transformation matrix of $A \circ B$ and $B \circ A$ with certain conditions

Let $A$ is 2x1 translation matrix and $B$ is 2x2 matrix of reflection or rotation matrix (reflection, rotation, etc.). Suppose I want to find the mapping of a $y=mx+c$ line and the mapping is done by ...
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Prove that the locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the circle $x^2+y^2-2ax=0$ is the circle $x^2+y^2-ax=0$.

Prove that the locus of the poles of tangents to the parabola $y^2=4ax$ with respect to the circle $x^2+y^2-2ax=0$ is the circle $x^2+y^2-ax=0$. I have encountered this question from SL Loney.I have ...
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Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other.

Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other. I tried to solve it but failed.Can someone please help me to ...