I have learnt that the equation of a tangent drawn ‘at’ a point $(x_1,y_1)$ to a circle $x^2 + y^2=a^2$ is $T=0$ that is $xx_1 + yy_1 - a^2=0$ where $T$ is obtained by replacing $x^2$ by $xx_1$ and $y^2$ by $yy_1$. Here we note that $x_1$ and $y_1$ are the coordinates of a point on the circle .
However the equation for pair of tangents drawn from a given point $P (x_1,y_1)$ to the circle is $T^2 =SS’$ . Here also $T$ is obtained in the same way except that we substitute the coordinates of the point that does not lie on the circle (i.e. we substitute the coordinates of the point from where the tangents are drawn) . Why is that $T$ changes by definition in the above two examples ? I understand that in the second example if we go by the definition of $T$ from the first example we’d have two equations , but why does $T$ change by definition ? Or is it just a convention ?