If `r_(1) and r_(2)` be the maximum and minimum radius of the circle which pass through the point (4, 3) and touch the circle `x^(2)+y^(2)=49`, then `(r_(1))/(r_(2))` is …….

Answer

To solve the problem, we need to find the ratio of the maximum radius \( r_1 \) and the minimum radius \( r_2 \) of circles that pass through the point \( (4, 3) \) and touch the circle given by the equation \( x^2 + y^2 = 49 \).

### Step-by-Step Solution:

1. **Identify the given circle**: The equation \( x^2 + y^2 = 49 \) represents a circle centered at the origin \( (0, 0) \) with a radius of \( 7 \) (since \( \sqrt{49} = 7 \)).

2. **Determine the distance from the point to the center of the given circle**: We need to find the distance from the point \( (4, 3) \) to the center \( (0, 0) \). Using the distance formula:
\[ d = \sqrt{(4 - 0)^2 + (3 - 0)^2} = \sqrt{16 + 9} = \sqrt{25} = 5. \]

3. **Calculate the maximum radius \( r_1 \)**: The maximum radius \( r_1 \) of the circle that passes through \( (4, 3) \) and touches the given circle is the distance from the point to the center of the given circle plus the radius of the given circle:
\[ r_1 = d + 7 = 5 + 7 = 12. \]

4. **Calculate the minimum radius \( r_2 \)**: The minimum radius \( r_2 \) of the circle that passes through \( (4, 3) \) and touches the given circle is the distance from the point to the center of the given circle minus the radius of the given circle:
\[ r_2 = d - 7 = 5 - 7 = -2. \]
Since a radius cannot be negative, we take the absolute value:
\[ r_2 = 2. \]

5. **Find the ratio \( \frac{r_1}{r_2} \)**: Now we can find the ratio of the maximum radius to the minimum radius:
\[ \frac{r_1}{r_2} = \frac{12}{2} = 6. \]

### Final Answer:
\[ \frac{r_1}{r_2} = 6. \]