Let C₁ and C₂ be two circles with C₂ lying inside C₁. A circle C lying inside C₁ touches C₁ internal

Q: Wait, why is the distance for internal tangency $R_1 - r$?

When a circle touches another internally, the smaller circle is contained within the larger one. Imagine the centers $P$ and $O_1$ and the point of tangency all lying on a single straight line. The distance from $O_1$ to the edge of $C_1$ is $R_1$. The distance from $P$ to the edge of $C_1$ (which is the same point of tangency) is $r$. Thus, the distance between the centers $PO_1$ must be the difference, $R_1 - r$.

Q: Why do we add the equations? Why not subtract?

We add them to eliminate the variable $r$. If we subtracted, we would be left with $r$ in our final expression, which doesn't help us find a fixed locus. Our goal is to find a relationship between the coordinates of $P$ and the fixed points $O_1$ and $O_2$ that is independent of the moving circle's size. Adding $PO_1 = R_1 - r$ and $PO_2 = R_2 + r$ yields $PO_1 + PO_2 = R_1 + R_2$, which is a constant.

Q: Is the locus always an ellipse?

Yes, provided the conditions of tangency are met. The sum of distances $PO_1 + PO_2$ is $R_1 + R_2$, which is a constant. This perfectly matches the geometric definition of an ellipse: the locus of a point such that the sum of its distances from two fixed points (the foci) is constant. Here, the foci are the centers $O_1$ and $O_2$.

Question

Let $C_1$ and $C_2$ be two circles with $C_2$ lying inside $C_1$. A circle $C$ lying inside $C_1$ touches $C_1$ internally and $C_2$ externally. Identify the locus of the centre of $C$.

Sigma Percentile · Accepted Answer

### The Geometry of Constraints

Imagine you are standing on a vast, flat plane. In front of you, there are two fixed circles: $C_1$ with center $O_1$ and radius $R_1$, and $C_2$ with center $O_2$ and radius $R_2$, where $C_2$ is nestled safely inside $C_1$.

Now, imagine a third circle, $C$, with center $P$ and radius $r$, dancing between them. This circle $C$ is not free; it is bound by two elegant rules: it touches $C_1$ internally and $C_2$ externally.

This is not just a problem of circles; it is a story of how constraints define a path.

### The Tangency Dance

To understand the path of $P$, we must first translate these physical constraints into the language of mathematics. When two circles touch internally, the distance between their centers is the difference of their radii.

So, for our moving circle $C$ and the large circle $C_1$, we have the distance equation:

$$PO_1 = R_1 - r$$

Now, consider the second constraint. When two circles touch externally, the distance between their centers is the sum of their radii. For our moving circle $C$ and the smaller circle $C_2$, we have:

$$PO_2 = R_2 + r$$

These two equations are the keys to the kingdom. We have a variable $r$—the radius of our moving circle—which changes as the circle slides around. We want the locus of $P$, a path that does not depend on $r$.

### The Elegant Elimination

This is where the beauty of algebra shines. We have two equations:

1) $PO_1 = R_1 - r$

2) $PO_2 = R_2 + r$

If we add these two equations together, something magical happens. The $-r$ from the first equation and the $+r$ from the second equation cancel each other out completely:

$$PO_1 + PO_2 = (R_1 - r) + (R_2 + r)$$

$$PO_1 + PO_2 = R_1 + R_2$$

Since $R_1$ and $R_2$ are the radii of our fixed circles, their sum $R_1 + R_2$ is a constant. We have arrived at a profound realization: the sum of the distances from the moving point $P$ to two fixed points $O_1$ and $O_2$ is constant.

### The Revelation: An Ellipse

Take a moment to appreciate this. In coordinate geometry, the definition of an ellipse is precisely the locus of a point such that the sum of its distances from two fixed points (the foci) is constant.

By simply adding our two tangency conditions, we have proven that the center $P$ of our moving circle must trace an **ellipse**.

The centers of the fixed circles, $O_1$ and $O_2$, act as the foci of this ellipse. The constant sum $R_1 + R_2$ represents the length of the major axis, $2a$.

You have successfully navigated the constraints, eliminated the variable, and uncovered the underlying geometric truth. This is the power of mathematical thinking: turning a complex, moving system into a static, elegant shape.

Animated Solution for Mathematics - Conic Sections: Let $C_{1}$ and $C_{2}$ be two circles with $C_{2}$ lying inside $C_{1}$ . A circle $C$ lying inside $C_{1}$ touches $C_{1}$ internally and $C_{2}$ externally. Identify the locus of the centre of $C$ .

Visualized Solution

The Fixed Circles $C_{1}$ and $C_{2}$

The Moving Circle $C$

Internal Tangency Condition

External Tangency Condition

Setting Up the Equations

Eliminating the Variable $r$

Simplifying the Expression

Geometric Definition of an Ellipse

Conclusion

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

The Geometry of Constraints

The Tangency Dance

The Elegant Elimination

The Revelation

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