This is a question from SL Loney's The Elements of Coordinate Geometry
Find the condition that the chord of contact of tangents from the point $$ (x' , y') $$ to the circle $$ x^2 + y^2 = a^2 $$ will subtend a right angle at the center.
My plan is first we should find the equation of chord of contact, which by definition is $$ xx' + yy' = a^2 $$ and then find the endpoints of this chord by solving the system of equations $$ xx' + yy' = a^2$$ $$x^2 + y^2 = a^2$$. Once we get the endpoints then the chord must satisfy the Pythagorean theorem in order to subtend a right angle 
That is to say it must follow this condition $$ a^2 + a^2 = AB^2 $$.
But the problem is by this approach I'm led into very messy algebra which seems unsolvable and hence I got stuck. I want to know is there any problem in my principles which I have mentioned above?
I looked up it's solution on internet and I found this 
Things become confusing when the author says " we make (1) homogenous with the help of (2)" My doubts are :-
- What is homogenous after all?
- Why do we need to that? Can't we solve it by geometrical rules which I have outlined above?
- Did he do that (that homogenous) only to solve this particular problem or is it something conventional which I'm ignorant of?
Thank you. Any help will be much appreciated.
$1)$ I bealive that making the equation $1$ homogeneous is substituting in itself the second equation, obtaining a polynomial equal to $0$.
$2)$You need that to solve, algebrically, the problem. You can solve this problem in a geometrical way: in fact so that angles $\measuredangle OAP= \measuredangle OBP=90$ then $OAPB$ has to be a square. From here the point $P$ stays on a circunference $\gamma$ of radius $OP=\sqrt{2}\cdot OB=\sqrt a \cdot OB$. The equation of $\gamma$ is so: $x^2+y^2=2a^2$.
$3)$ Making an equation homogeneous can be sometimes a really helpful tecnique to solve algebrically or geometrically a problem.