Find the locus of intersection of 2 circles(equal) which are described on 2 sides EF and EG of a triangles as chords in a rectangular hyperbola whose centre is middle point of FG and which passes through E,F,G

To find the locus of intersection of two equal circles described on the sides EF and EG of a triangle, where these circles act as chords of a rectangular hyperbola centered at the midpoint of FG and passing through points E, F, and G, we need to break down the problem step by step.

Understanding the Geometry

Let's start by visualizing the scenario. We have a triangle EFG, and we are interested in the sides EF and EG. The midpoint of FG will be our center for the circles we are considering. Since the circles are equal and described on the sides of the triangle, they will have the same radius.

Defining the Circles

Let’s denote the midpoint of FG as M. The circles can be defined as follows:

• Circle 1 (C1) is centered at M and has a radius equal to the length of EF.
• Circle 2 (C2) is also centered at M and has a radius equal to the length of EG.

Since the circles are equal, they will intersect at points that are equidistant from M along the line that bisects the angle at E between sides EF and EG.

Finding the Intersection Points

The intersection points of the two circles can be found using the general equation of a circle. The equation for both circles can be expressed as:

• For Circle 1: (x - Mx)² + (y - My)² = r²
• For Circle 2: (x - Mx)² + (y - My)² = r²

Here, (Mx, My) are the coordinates of the midpoint M, and r is the radius of the circles. Since both circles are equal, their equations will be identical, and the intersection points will lie along the line that connects the points where the circles intersect.

Using the Hyperbola

Now, since the circles are chords of a rectangular hyperbola, we need to consider the properties of the hyperbola. A rectangular hyperbola has the property that the product of the distances from any point on the hyperbola to the two foci is constant. The foci of the hyperbola can be determined based on the coordinates of points E, F, and G.

Determining the Locus

The locus of the intersection points of the two circles will form a curve. Since the circles are equal and symmetric about the line bisecting angle E, the locus will also be symmetric. The intersection points will trace a path that can be described by the equation of the hyperbola.

Final Equation

To derive the specific equation of the locus, we can use the coordinates of points E, F, and G, along with the properties of the hyperbola. The general form of the rectangular hyperbola can be expressed as:

• (x²/a²) - (y²/b²) = 1

Here, a and b are constants that can be determined based on the distances between points E, F, and G. The intersection points of the circles will lie on this hyperbola, thus forming the locus we are looking for.

Conclusion

In summary, the locus of intersection of the two equal circles described on the sides EF and EG of triangle EFG, which are chords of a rectangular hyperbola centered at the midpoint of FG, will trace out a path that can be represented by the equation of the hyperbola itself. This geometric relationship beautifully illustrates the interplay between circles and hyperbolas in coordinate geometry.