Animated Solution for Mathematics - Circles: Through a fixed point $(h,k)$ secants are drawn to the circle $x^2+y^2=r^2$. Show that the locus of the mid-points of the secants intercepted by the circle is $x^2+y^2=hx+ky$.

Imagine a circle centered at the origin defined by the equation $x^2+y^2=r^2$. We are given a fixed point $P(h,k)$ in the plane from which we draw various secant lines.

Each secant line intersects the circle to form a chord. Our objective is to determine the locus of the midpoints $(x_1,y_1)$ of all such chords that pass through the fixed point $P$.

When dealing with chords and midpoints, the most efficient tool is the $T=S_1$ theorem. For any circle $S=x^2+y^2-r^2=0$, the equation of a chord with a given midpoint $(x_1,y_1)$ is expressed as:

In this expression, $T$ represents the tangent-like form $x{x}_1+y{y}_1-r^2$, and $S_1$ is the value of the circle's equation evaluated at the midpoint, given by $x_1^2+y_1^2-r^2$. This theorem allows us to bypass the complex algebra of calculating intersection points directly.

Applying this to our circle $x^2+y^2-r^2=0$, we substitute the midpoint $(x_1,y_1)$ into the $T=S_1$ framework:

Observing the equation, the $-r^2$ terms on both sides cancel out. This leaves us with a simplified linear relationship for the chord:

Because every secant line must pass through the fixed point $P(h,k)$, the coordinates of $P$ must satisfy the equation of the chord. We substitute $x=h$ and $y=k$ into our simplified chord equation:

This is the governing condition that every midpoint $(x_1,y_1)$ must satisfy. To define the locus, we replace the specific coordinates $(x_1,y_1)$ with the general variables $(x,y)$.

The resulting equation for the locus is:

This is the equation of a circle. It represents a circle that passes through both the origin $(0,0)$ and the fixed point $P(h,k)$.

The line segment joining the origin and $P$ serves as the diameter of this new circle. Thus, the midpoints of all secants drawn from a fixed point trace out a perfect circle.