Animated Solution for Mathematics - Circles: If a tangent to the circle $x^2+y^2=1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is

Imagine you are standing in a coordinate plane. At the center, there sits a perfect, serene unit circle, defined by the equation $x^2+y^2=1$.

Now, imagine a line—a tangent—that kisses this circle at a single, precise point $T(x_1,y_1)$. As this line rotates around the circle, it carves out intercepts on the $x$ and $y$ axes, which we label as points $P$ and $Q$.

Our goal is to track the heartbeat of this system: the midpoint $M(h,k)$ of the segment $PQ$. As the tangent dances around the circle, we seek the path traced by this midpoint.

To begin, we must pin down the equation of our moving tangent. For any point $(x_1,y_1)$ on the circle $x^2+y^2=1$, the equation of the tangent line is given by:

To find where this line hits the axes, we set one coordinate to zero. For the $x$-intercept $P$, we set $y=0$, yielding $x=\frac{1}{x_1}$. Thus, $P=(\frac{1}{x_1},0)$.

Similarly, for the $y$-intercept $Q$, we set $x=0$, yielding $y=\frac{1}{y_1}$. Thus, $Q=(0,\frac{1}{y_1})$.

Now, we define our midpoint $M(h,k)$. By the midpoint formula, we have:

These equations bridge the tangent's position and the midpoint's location. Rearranging these, we express the original tangency point $(x_1,y_1)$ in terms of our midpoint coordinates:

The point $(x_1,y_1)$ is a prisoner of the circle and must satisfy the constraint $x_1^2+y_1^2=1$. This constraint dictates the entire motion of the midpoint.

By substituting our expressions for $x_1$ and $y_1$ into this constraint, we obtain:

Expanding the squares, we get:

Multiplying the entire equation by $4$, we arrive at:

Taking the common denominator, we have $\frac{k^2+h^2}{h^2{k}^2}=4$. Cross-multiplying yields $h^2+k^2=4h^2{k}^2$.

Finally, replacing $(h,k)$ with the general coordinates $(x,y)$, we reach our destination:

This is the equation of the locus. You have successfully mapped the path, turning a dynamic, moving system into a static, elegant expression of geometry.