STATEMENT-1 : The agnle between the tangents drawn from the point (6, 8) to the circle `x^(2) + y^(2) = 50` is `90^(@)`.
and
STATEMENT-2 : The locus of point of intersection of perpendicular tangents to the circle `x^(2) + y^(2) = r^(2)` is `x^(2) + y^(2) = 2r^(2)`.

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 given in the question.

### Step 1: Analyze Statement 1
We need to determine if the angle between the tangents drawn from the point (6, 8) to the circle defined by the equation \(x^2 + y^2 = 50\) is \(90^\circ\).

1. **Identify the Circle's Center and Radius**:
- The given circle is \(x^2 + y^2 = 50\).
- The center of the circle is at \((0, 0)\) and the radius \(r\) is \(\sqrt{50} = 5\sqrt{2}\).

2. **Calculate the Power of the Point**:
- The power of a point \(P(x_1, y_1)\) with respect to a circle is given by:
\[ \text{Power} = x_1^2 + y_1^2 - r^2 \]
- For the point \(P(6, 8)\):
\[ \text{Power} = 6^2 + 8^2 - 50 = 36 + 64 - 50 = 50 \]

3. **Length of Tangents**:
- The length of the tangents \(L\) from the point \(P\) to the circle is given by:
\[ L = \sqrt{\text{Power}} = \sqrt{50} = 5\sqrt{2} \]

4. **Using the Tangent Angle Formula**:
- The angle \(\theta\) between the two tangents can be calculated using the formula:
\[ \tan \theta = \frac{2LR}{L^2 - R^2} \]
- Here, \(L = R = 5\sqrt{2}\):
\[ L^2 = (5\sqrt{2})^2 = 50 \]
- Therefore, the denominator becomes:
\[ L^2 - R^2 = 50 - 50 = 0 \]
- Since the denominator is zero, \(\tan \theta\) approaches infinity, which implies:
\[ \theta = 90^\circ \]

### Conclusion for Statement 1:
Thus, Statement 1 is **True**.

### Step 2: Analyze Statement 2
We need to verify if the locus of the point of intersection of perpendicular tangents to the circle \(x^2 + y^2 = r^2\) is given by \(x^2 + y^2 = 2r^2\).

1. **Understanding the Geometry**:
- The tangents to the circle from a point \(P\) intersect at a right angle (90 degrees).
- The center of the circle is at the origin \((0, 0)\).

2. **Finding the Locus**:
- Let the point of intersection of the tangents be \(Q(x, y)\).
- The distance from the center to the point \(Q\) must equal the radius \(r\) of the circle, and since the tangents are perpendicular, we can use the relationship:
\[ OQ^2 = OP^2 + PQ^2 \]
- The distance \(OQ\) is the radius of the circle, and \(OP\) is the distance from the center to the point from which the tangents are drawn.

3. **Using the Right Triangle**:
- Since the tangents are perpendicular, we can derive that:
\[ OQ^2 = OP^2 + OP^2 = 2(OP^2) \]
- This gives us:
\[ x^2 + y^2 = 2r^2 \]

### Conclusion for Statement 2:
Thus, Statement 2 is also **True**.

### Final Conclusion:
Both statements are true.