find the locus of middle points of chords of ellipse which are drawn through positive end of minor axis.

To find the locus of the midpoints of chords of an ellipse that pass through the positive end of the minor axis, we first need to establish the equation of the ellipse and the coordinates of the points involved. Let's consider the standard form of the ellipse centered at the origin, which is given by:

The Equation of the Ellipse

The equation of an ellipse with semi-major axis \(a\) and semi-minor axis \(b\) is:

\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

For our purposes, the positive end of the minor axis is the point \((0, b)\). We will denote the coordinates of the midpoint of a chord that passes through this point as \((h, k)\).

Finding the Chord Equation

Any chord of the ellipse can be represented in terms of its endpoints. If we denote the endpoints of the chord as \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \((h, k)\) can be expressed as:

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

Since the chord passes through the point \((0, b)\), we can use the property of the ellipse to derive a relationship between the coordinates of the endpoints and the midpoint.

Using the Ellipse Property

For any point \((x, y)\) on the ellipse, it satisfies the ellipse equation. Therefore, both endpoints of the chord must satisfy:

\(\frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} = 1\)

\(\frac{x_2^2}{a^2} + \frac{y_2^2}{b^2} = 1\)

Now, substituting \(y_1\) and \(y_2\) in terms of \(h\) and \(k\) gives us a way to express the relationship between the midpoints and the endpoints.

Deriving the Locus

To find the locus of midpoints, we can use the fact that the midpoint of a chord that passes through a fixed point can be expressed in terms of the coordinates of that fixed point. The equation of the chord can be derived using the slope-intercept form or parametric equations. However, a more straightforward approach is to use the property of the ellipse:

The locus of the midpoints of all chords passing through a fixed point on the ellipse is another ellipse. Specifically, if we take the point \((0, b)\) as our fixed point, the locus of midpoints will be an ellipse with the same center but scaled down.

Final Equation of the Locus

The locus of the midpoints of the chords through the point \((0, b)\) can be derived to be:

\(\frac{h^2}{a^2/4} + \frac{k^2}{b^2/4} = 1\)

This indicates that the locus is an ellipse with semi-major axis \(a/2\) and semi-minor axis \(b/2\), centered at the origin.

Summary

In summary, the locus of the midpoints of chords of the ellipse that pass through the positive end of the minor axis \((0, b)\) is another ellipse, specifically:

\(\frac{h^2}{a^2/4} + \frac{k^2}{b^2/4} = 1\)

This result highlights the beautiful symmetry and properties of ellipses in geometry.