substituting a point inside the circle in the equation $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c=0$

Question

I have learned that when I substitute a point outside the circle to the equation $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c=0$ gives the equation of chord of contact and when I substitute a point on the circle, I get the equation of the tangent at that point. But what happens when I substitute a point inside a circle in this equation (including the Centre). does it give any meaningful geometrical representation

Answer 1

Algebraically, being given a circle with equation :

$$x^2+y^2+2fx+2gy+c=0$$

the correspondence:

$$\underbrace{(x_1,y_1)}_{\text{point}} \ \leftrightarrow \ \underbrace{xx_1+yy_1+f(x+x_1)+g(y+y_1)+c=0}_{\text{line}}$$

is called the pole $ \ \leftrightarrow \ $ polar line correspondence.

For the case of an interior point, a geometrical construction exists but is less evident than when the point is outside the circle. Here is a construction of the polar line of point $M$ in this case.

Take two chords $AB$ and $A'B'$ passing through point $M$. Construct their tangents to the circle : their respective intersections $C$ and $C'$ belong to the polar line ; otherwise said, the polar line is the red line obtained by joining $C$ and $C'$.

Remark 1: please note that the line issued from the origin and passing by $M$ is orthogonal to the polar line of $M$.

Remar 2: In both cases (point $M$ outside or inside the circle), we use tangents "issued from" or "intersecting in" a certain point. You will certainly see sooner or later that it is one of the manifestations of a very important concept, i.e., duality.