Exploring Locus Problems in Math Olympiad Geometry

Welcome to a thrilling exploration of locus problems in geometry, a crucial concept for anyone preparing for math competitions like the IOQM, American Math Competition, and GMD. Whether you're aiming for ISI, CMI, or just looking to sharpen your mathematical skills, understanding loci will give you a solid edge.

What is a Locus?

In simpler terms, a locus is the path traced by a moving point that follows a specific rule.Imagine you have a fixed point, O, and a moving point, P. Point P doesn't move randomly; it follows a specific rule. Our goal is to find out the path that point P traces as it moves according to this rule. This path is known as the locus of point P.

Example 1: Drawing a Circle

Let’s start with a simple rule: point P is always 3 units away from point O. What happens then? P traces out a circle!

  1. Dynamic View: Picture point P moving around point O, always keeping a distance of 3 units. As P moves, a circle forms. This helps you see how the circle is created step-by-step.
  2. Static View: Think of the circle as all points that are 3 units away from O. This gives you a complete picture of the circle at once.

Both views are important and help you understand the circle in different ways.

Example 2: Creating an Ellipse

Next, let’s try a different rule. Suppose you have two fixed points, O1 and O2, and a moving point, P. The rule is that the sum of the distances from P to O1 and O2 is always 5 units. What shape does P trace out? The answer is an ellipse!

To understand this, imagine P moving so that the distances to O1 and O2 always add up to 5. Visualizing this movement helps you see how the ellipse forms.

Example 3: Rolling Circles

For our final example, imagine a big fixed circle with a diameter of 4 cm and a small moving circle with a diameter of 1 cm. If the small circle rolls around the big circle, what path does a point on the edge of the small circle trace? This path is called a hypocycloid.

As the small circle rolls, the point on its edge creates a unique and interesting path. Visualizing this helps you understand the movement and the resulting shape.

Try It Yourself!

You have two fixed points, A and B, and a moving point, P. The sum of the distances from P to A and B is constant. What path does P trace out?

Share your answers in the comments!

How to Solve Locus Problems | Math Olympiad Geometry Concept | Cheenta

I.S.I Entrance Solution - locus of a moving point

This is an I.S.I. Entrance Solution

Problem:

P is a variable point on a circle C and Q is a fixed point on the outside of C. R is a point in PQ dividing it in the ratio p:q, where p> 0 and q > 0 are fixed. Then the locus of R is

(A) a circle;  (B) an ellipse;  (C) a circle if p = q and an ellipse otherwise; (C) none of the above curves;

Discussion:

Also see I.S.I. & C.M.I. Entrance Program

WLOG C be a unit circle centered at (0,0). Then ( P= (\cos(t), \sin (t)) ). Suppose Q = (a, b).

If R divides PQ in m: n ratio, then ( R = \left( \frac{ma + ncos(t)}{m+n} , \frac{mb+ nsin(t)}{m+n} \right) )

Then the distance of R from ( \left ( \frac{ma}{m+n}, \frac{mb}{m+n} \right) ) is ( \sqrt { \left( \frac{ma + ncos(t)}{m+n} - \frac{ma}{m+n} \right)^2 + \left( \frac{mb +nsin(t)}{m+n} - \frac{mb}{m+n} \right)^2}  )

But this equals: ( \sqrt {\frac{m^2+ n^2} {(m+n)^2}} ) which is a constant. Hence R is at a constant distance from ( \left ( \frac{ma}{m+n}, \frac{mb}{m+n} \right) )  is a constant.

Hence R traces out a circle.

Theoretical remark:

The parametric equation of a circle with unit radius centered at origin is ( \cos(t), \sin(t) ) . This is the key idea in this problem.

Another idea is: if PQ is divided in m : n ratio R, what is the coordinate of R. You will need 'section formula' to compute that (this formula is a consequence of similarity of triangles).