Find the maximum length of chord of the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` then find the locus of midpoint of PQ

Explanation

To find the maximum length of the chord of the ellipse given by the equation

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \]

we will follow these steps:

### Step 1: General Points on the Ellipse
Let \( P \) and \( Q \) be two points on the ellipse. We can express these points in terms of a parameter \( \theta \):
- For point \( P \):
\[ P = (a \cos \theta, b \sin \theta) \]
- For point \( Q \):
\[ Q = (a \cos(\theta + \phi), b \sin(\theta + \phi)) \]
where \( \phi \) is the angle between the two points.

### Step 2: Length of Chord PQ
The length of the chord \( PQ \) can be calculated using the distance formula:
\[ PQ = \sqrt{(a \cos(\theta + \phi) - a \cos \theta)^2 + (b \sin(\theta + \phi) - b \sin \theta)^2} \]

### Step 3: Simplifying the Length Expression
Using the cosine and sine addition formulas:
\[ \cos(\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi \]
\[ \sin(\theta + \phi) = \sin \theta \cos \phi + \cos \theta \sin \phi \]
we can substitute these into the distance formula and simplify the expression.

### Step 4: Finding Maximum Length
To find the maximum length of the chord, we need to maximize the expression derived in Step 3. The maximum length occurs when the angle \( \phi \) is such that the sine function achieves its maximum value.

### Step 5: Maximum Length of the Chord
After simplification, we find that the maximum length \( L \) of the chord is given by:
\[ L = \sqrt{a^2 + b^2} \]
This represents the maximum length of the chord of the ellipse.

### Step 6: Locus of the Midpoint of Chord PQ
The midpoint \( M \) of the chord \( PQ \) can be expressed as:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points \( P \) and \( Q \) into this formula, we can derive the locus of the midpoint.

### Step 7: Locus Equation
The locus of the midpoint \( M \) will be an ellipse given by the equation:
\[ \frac{x^2}{\left(\frac{a}{2}\right)^2} + \frac{y^2}{\left(\frac{b}{2}\right)^2} = 1 \]

### Summary
1. The maximum length of the chord of the ellipse is \( \sqrt{a^2 + b^2} \).
2. The locus of the midpoint of the chord \( PQ \) is given by the equation of an ellipse with semi-major axis \( \frac{a}{2} \) and semi-minor axis \( \frac{b}{2} \).