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 maximum and minimum radii of circles that pass through the point (4, 3) and touch the circle defined 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 with center at the origin (0, 0) and radius \(r = 7\).

2. **Determine the Position of the Point (4, 3)**:
To find out where the point (4, 3) lies in relation to the given circle, we substitute \(x = 4\) and \(y = 3\) into the circle's equation:
\[ 4^2 + 3^2 = 16 + 9 = 25 \]
Since \(25 < 49\), the point (4, 3) lies inside the circle.

3. **Understanding the Configuration**:
We need to find two circles that pass through the point (4, 3) and touch the given circle. The maximum radius circle will be outside the given circle, and the minimum radius circle will be inside the given circle.

4. **Finding the Distance from the Center to the Point**:
Calculate the distance from the center of the circle (0, 0) to the point (4, 3):
\[ d = \sqrt{(4 - 0)^2 + (3 - 0)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

5. **Finding the Minimum Radius \(r_1\)**:
The minimum radius \(r_1\) is the distance from the point (4, 3) to the point where the circle touches the given circle:
\[ r_1 = 7 - d = 7 - 5 = 2 \]

6. **Finding the Maximum Radius \(r_2\)**:
The maximum radius \(r_2\) is the distance from the point (4, 3) to the outer edge of the given circle:
\[ r_2 = d + 7 = 5 + 7 = 12 \]

7. **Calculating the Ratio \( \frac{r_1}{r_2} \)**:
Now, we can find the ratio of the maximum radius to the minimum radius:
\[ \frac{r_1}{r_2} = \frac{2}{12} = \frac{1}{6} \]

### Final Answer:
\[ \frac{r_1}{r_2} = \frac{1}{6} \]