Tangents are drawn from the point (17, 7) to the circle `x^2+y^2=169`, Statement I The tangents are mutually perpendicular Statement, ll The locus of the points frorn which mutually perpendicular tangents can be drawn to the given circle is `x^2 +y^2=338`

Options

A. Statement I is True, statement II is true, Statement II is a correct explanation for Statement I
B. Statement I is true, Statement II is true, Statement II is not a correct explanation for Statement I
C. Statement I is true, Statement II is false
D. Statement I is false, Statement II is true

Answer

To solve the problem step by step, we will analyze the statements given and verify their correctness.

### Step 1: Understand the Circle
The equation of the circle is given as:
\[ x^2 + y^2 = 169 \]
This represents a circle centered at the origin (0, 0) with a radius \( r = \sqrt{169} = 13 \).

**Hint:** Identify the center and radius of the circle from its equation.

### Step 2: Determine the Point from Which Tangents are Drawn
We are drawing tangents from the point \( (17, 7) \) to the circle.

**Hint:** Note the coordinates of the external point from which the tangents are being drawn.

### Step 3: Check if the Tangents are Perpendicular
For two tangents drawn from an external point to a circle to be mutually perpendicular, the following condition must hold:
\[ OP^2 = r^2 + r^2 \]
where \( OP \) is the distance from the center of the circle to the external point, and \( r \) is the radius of the circle.

1. Calculate \( OP \):
\[ OP = \sqrt{(17 - 0)^2 + (7 - 0)^2} = \sqrt{17^2 + 7^2} = \sqrt{289 + 49} = \sqrt{338} \]

2. Now, check if \( OP^2 = 2r^2 \):
\[ OP^2 = 338 \quad \text{and} \quad r^2 = 169 \]
\[ 2r^2 = 2 \times 169 = 338 \]

Since \( OP^2 = 2r^2 \), the tangents are mutually perpendicular.

**Hint:** Use the distance formula to find the distance from the center to the point and compare it with the condition for perpendicular tangents.

### Step 4: Determine the Locus of Points from Which Perpendicular Tangents Can Be Drawn
The locus of points from which mutually perpendicular tangents can be drawn to a circle is given by:
\[ x^2 + y^2 = 2r^2 \]
Substituting \( r^2 = 169 \):
\[ x^2 + y^2 = 2 \times 169 = 338 \]

**Hint:** Recall the formula for the locus of points that can draw perpendicular tangents to a circle.

### Conclusion
- **Statement I**: The tangents are mutually perpendicular. (True)
- **Statement II**: The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is \( x^2 + y^2 = 338 \). (True)

Both statements are correct.

**Final Answer: Both statements are true.**