If the chord of contact of the tangents from the point `(alpha, beta)` to the circle `x^(2)+y^(2)=r_(1)^(2)` is a tangent to the circle `(x-a)^(2)+(y-b)^(2)=r_(2)^(2)`, then

Options

A. `r_(2)^(2)(alpha^(2)+beta^(2))=(r_(1)^(2)- a alpha - b beta)^(2)`
B. `r_(2)^(2)(alpha^(2)+beta^(2))=(r_(1)^(2)+a alpha=b beta)^(2)`
C. `r_(2)^(2)(alpha^(2)+beta^(2))=(r_(1)^(2)-a alpha +b beta)^(2)`
D. `r_(2)^(2)(alpha^(2)+beta^(2))=(r_(1)^(2)+a alpha+b beta)^(2)`

Answer

To solve the problem, we need to derive the relationship between the coordinates of the point from which the tangents are drawn, the radius of the first circle, and the parameters of the second circle.

### Step-by-Step Solution:

1. **Identify the Circle and Point**:
The first circle is given by the equation \( x^2 + y^2 = r_1^2 \) with center at \( (0, 0) \) and radius \( r_1 \). The point from which tangents are drawn is \( ( \alpha, \beta ) \).

2. **Equation of the Chord of Contact**:
The chord of contact from the point \( ( \alpha, \beta ) \) to the first circle can be derived using the formula:
\[ x \alpha + y \beta = r_1^2 \]
This represents the line that is tangent to the first circle.

3. **Second Circle**:
The second circle is given by the equation \( (x - a)^2 + (y - b)^2 = r_2^2 \) with center at \( (a, b) \) and radius \( r_2 \).

4. **Condition for Tangency**:
The chord of contact (line) must be tangent to the second circle. The condition for a line \( Ax + By + C = 0 \) to be tangent to a circle \( (x - h)^2 + (y - k)^2 = r^2 \) is given by:
\[ \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}} = r \]
For our case, the line can be rewritten as:
\[ \alpha x + \beta y - r_1^2 = 0 \]
Here, \( A = \alpha \), \( B = \beta \), and \( C = -r_1^2 \).

5. **Distance from Center of Second Circle**:
The distance from the center \( (a, b) \) of the second circle to the line is:
\[ \frac{|\alpha a + \beta b - r_1^2|}{\sqrt{\alpha^2 + \beta^2}} \]
This distance must equal the radius \( r_2 \) of the second circle:
\[ \frac{|\alpha a + \beta b - r_1^2|}{\sqrt{\alpha^2 + \beta^2}} = r_2 \]

6. **Squaring Both Sides**:
To eliminate the absolute value and the square root, we square both sides:
\[ (\alpha a + \beta b - r_1^2)^2 = r_2^2 (\alpha^2 + \beta^2) \]

7. **Final Relationship**:
This gives us the final relationship between \( r_1, r_2, \alpha, \beta, a, \) and \( b \):
\[ (\alpha a + \beta b - r_1^2)^2 = r_2^2 (\alpha^2 + \beta^2) \]