If x,y `in`R and satisfy `(x+5)^2+(y-12)^2=14^2` then the minimum value of `x^2+y^2` is

Options

A. 2
B. 1
C. `sqrt(3)`
D. `sqrt(2)`

Answer

To find the minimum value of \( x^2 + y^2 \) given the equation \( (x + 5)^2 + (y - 12)^2 = 14^2 \), we can follow these steps:

### Step 1: Understand the given equation
The equation \( (x + 5)^2 + (y - 12)^2 = 14^2 \) represents a circle in the Cartesian plane. The center of the circle is at the point \( (-5, 12) \) and the radius is \( 14 \).

### Step 2: Identify the distance from the origin
We need to find the minimum distance from the origin \( (0, 0) \) to any point on the circle. The distance \( d \) from the center of the circle \( (-5, 12) \) to the origin can be calculated using the distance formula:
\[ d = \sqrt{(-5 - 0)^2 + (12 - 0)^2} = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13. \]

### Step 3: Determine the minimum distance to the circle
The minimum distance from the origin to the circle is the distance from the origin to the center of the circle minus the radius of the circle:
\[ \text{Minimum distance} = d - \text{radius} = 13 - 14 = -1. \]
Since the distance cannot be negative, this indicates that the origin is inside the circle.

### Step 4: Calculate the minimum value of \( x^2 + y^2 \)
The minimum value of \( x^2 + y^2 \) occurs when we are at the closest point on the circle to the origin. The closest point on the circle will be at a distance of \( 1 \) from the origin (since the radius is \( 14 \) and the distance to the center is \( 13 \)):
\[ \text{Minimum distance from the origin to the circle} = 1. \]

### Step 5: Find the minimum value of \( x^2 + y^2 \)
The minimum value of \( x^2 + y^2 \) is the square of the minimum distance:
\[ \text{Minimum value of } x^2 + y^2 = (1)^2 = 1. \]

### Final Answer
Thus, the minimum value of \( x^2 + y^2 \) is \( \boxed{1} \).

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