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

Answer

To find the locus of the midpoints of chords of an ellipse drawn through the positive end of the minor axis, we can follow these steps:

### Step 1: Define the Ellipse
The standard form of the ellipse is given by the equation:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

### Step 2: Identify the Coordinates of the Minor Axis
The minor axis of the ellipse is vertical, and its endpoints are at:
\[ (0, b) \quad \text{and} \quad (0, -b) \]
The positive end of the minor axis is at the point \( (0, b) \).

### Step 3: General Point on the Ellipse
A general point \(A\) on the ellipse can be represented as:
\[ A(a \cos \theta, b \sin \theta) \]
where \(\theta\) is a parameter.

### Step 4: Midpoint of the Chord
Let \(B\) be the point on the ellipse through which the chord is drawn. The coordinates of \(B\) can also be expressed as \(B(a \cos \phi, b \sin \phi)\). The midpoint \(M\) of the chord \(AB\) can be calculated as:
\[ M\left(\frac{0 + a \cos \theta}{2}, \frac{b + b \sin \theta}{2}\right) = \left(\frac{a \cos \theta}{2}, \frac{b(1 + \sin \theta)}{2}\right) \]
Let \(M\) have coordinates \((h, k)\):
\[ h = \frac{a \cos \theta}{2} \quad \text{and} \quad k = \frac{b(1 + \sin \theta)}{2} \]

### Step 5: Express \(\cos \theta\) and \(\sin \theta\) in terms of \(h\) and \(k\)
From the equations for \(h\) and \(k\), we can express \(\cos \theta\) and \(\sin \theta\):
1. From \(h\):
\[ \cos \theta = \frac{2h}{a} \]
2. From \(k\):
\[ 2k = b(1 + \sin \theta) \implies \sin \theta = \frac{2k}{b} - 1 \]

### Step 6: Use the Pythagorean Identity
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ \left(\frac{2k}{b} - 1\right)^2 + \left(\frac{2h}{a}\right)^2 = 1 \]

### Step 7: Simplify the Equation
Expanding the equation:
\[ \left(\frac{2k}{b} - 1\right)^2 = \frac{4k^2}{b^2} - \frac{4k}{b} + 1 \]
Thus, the equation becomes:
\[ \frac{4k^2}{b^2} - \frac{4k}{b} + 1 + \frac{4h^2}{a^2} = 1 \]
Simplifying gives:
\[ \frac{4k^2}{b^2} + \frac{4h^2}{a^2} - \frac{4k}{b} = 0 \]

### Step 8: Rearranging to Find the Locus
This can be rearranged to find the locus of the midpoints:
\[ \frac{4k^2}{b^2} + \frac{4h^2}{a^2} = \frac{4k}{b} \]
This is the equation of a parabola or a conic section representing the locus of the midpoints of the chords.

### Final Result
The locus of the midpoints of the chords of the ellipse drawn through the positive end of the minor axis is given by the equation derived above.