Find the locus of the middle points of the chords of the circle x²+y²+2gx+2fy+c=0 which subtend a right angle at the origin.

To find the locus of the midpoints of the chords of the circle defined by the equation \(x^2 + y^2 + 2gx + 2fy + c = 0\) that subtend a right angle at the origin, we can follow a systematic approach. This involves using some properties of circles and geometry. Let's break it down step by step.

Understanding the Circle Equation

The given equation of the circle can be rewritten in standard form. The general form \(x^2 + y^2 + 2gx + 2fy + c = 0\) can be rearranged to identify the center and radius:

• Center: \((-g, -f)\)
• Radius: \(r = \sqrt{g^2 + f^2 - c}\)

Chords Subtending a Right Angle

For a chord to subtend a right angle at the origin, a specific geometric property comes into play: the midpoint of such a chord lies on a circle whose diameter is the line segment from the origin to the circle's center. This is a consequence of the fact that the angle subtended by a diameter at any point on the circle is a right angle.

Finding the Midpoint

Let’s denote the midpoint of the chord as \(M(h, k)\). The endpoints of the chord can be represented as \(A(x_1, y_1)\) and \(B(x_2, y_2)\). The midpoint \(M\) can be expressed as:

• \(h = \frac{x_1 + x_2}{2}\)
• \(k = \frac{y_1 + y_2}{2}\)

Using the Right Angle Condition

Since the chord subtends a right angle at the origin, we can use the property of the dot product. The vectors \(OA\) and \(OB\) (where \(O\) is the origin) must satisfy:

\(x_1 x_2 + y_1 y_2 = 0\)

This means that the product of the coordinates of the endpoints of the chord must equal zero, indicating that they are perpendicular.

Substituting into the Circle Equation

Now, substituting \(x_1\) and \(y_1\) in terms of \(h\) and \(k\) into the circle's equation gives:

\(h^2 + k^2 + 2g(h) + 2f(k) + c = 0\)

From the right angle condition, we can express \(x_2\) and \(y_2\) in terms of \(h\) and \(k\) as well. However, we need to find a relationship that eliminates \(x_1\) and \(y_1\) in favor of \(h\) and \(k\).

Deriving the Locus Equation

Using the midpoint formula and the right angle condition, we can derive the locus of the midpoints. The locus of the midpoints of the chords that subtend a right angle at the origin can be shown to be a circle centered at the point \((-g, -f)\) with a radius that can be derived from the original circle's properties.

Final Locus Equation

The locus of the midpoints \(M(h, k)\) can be expressed as:

\(h^2 + k^2 + gh + fk = \frac{g^2 + f^2 - c}{2}\)

This represents a circle in the \(hk\)-plane, centered at \((-g/2, -f/2)\) with a radius that can be calculated based on the original circle's parameters.

In summary, the locus of the midpoints of the chords of the given circle that subtend a right angle at the origin is another circle, and this relationship beautifully illustrates the interplay between geometry and algebra in coordinate systems.