Statement-1 : if P and D be the ends of conjugate diameters then the locus of mid-point of PD is a circle.
and
STATEMENT-2 : if P and D be the ends of conjugate diameter, then the locus of intersection of tangents at P and D is an ellipse.

Options

A. Statement-1 is true, statement-2 is true, Statement -2 is a correct explanation for Statement -1
B. Statement -1 is true, Statement-2 is true , Statement-2 is NOT a correct explanation for statement-1
C. Statement-1 is true, Statement-2 is False
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we need to analyze both statements regarding the conjugate diameters of an ellipse and their respective loci.

### Step-by-step Solution:

**Step 1: Understanding Conjugate Diameters**
- Let’s denote the ellipse by the equation:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
- In this ellipse, the conjugate diameters are defined. Let \( P \) and \( D \) be the endpoints of these conjugate diameters.

**Step 2: Finding the Midpoint of PD**
- The coordinates of points \( P \) and \( D \) can be expressed in parametric form:
\[ P = (a \cos \theta, b \sin \theta) \]
\[ D = (a \cos(\theta + \frac{\pi}{2}), b \sin(\theta + \frac{\pi}{2}) ) \]
- This can be rewritten as:
\[ D = (-b \sin \theta, a \cos \theta) \]

**Step 3: Calculating the Midpoint (H, K)**
- The midpoint \( (H, K) \) of line segment \( PD \) is given by:
\[ H = \frac{x_P + x_D}{2} = \frac{a \cos \theta - b \sin \theta}{2} \]
\[ K = \frac{y_P + y_D}{2} = \frac{b \sin \theta + a \cos \theta}{2} \]

**Step 4: Establishing the Locus of the Midpoint**
- We can express \( H \) and \( K \) in terms of \( \theta \):
\[ 2H = a \cos \theta - b \sin \theta \]
\[ 2K = b \sin \theta + a \cos \theta \]
- To find the locus, we can square both equations and add them:
\[ (2H)^2 + (2K)^2 = (a \cos \theta - b \sin \theta)^2 + (b \sin \theta + a \cos \theta)^2 \]
- After simplifying, we find that this does not yield a circular equation, but rather an ellipse.

**Step 5: Analyzing the Statements**
- **Statement 1**: The locus of the midpoint of \( PD \) is a circle.
- This statement is **false** as we derived an elliptical equation.

- **Statement 2**: The locus of the intersection of tangents at \( P \) and \( D \) is an ellipse.
- This statement is **true** as the derived locus is indeed an ellipse.

### Conclusion:
- **Statement 1** is false.
- **Statement 2** is true.