Two circle pass through `(-1,4)` and their centres lie on `x^2+y^2+2x+4y=4` . If `r_1 and r_2` are maximum and minimum radii and `r_1/r_2=a+bsqrt2` then value of a+b is

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the given equation of the circle
The centers of the two circles lie on the circle defined by the equation:
\[ x^2 + y^2 + 2x + 4y = 4 \]
We can rewrite this equation in standard form by completing the square.

### Step 2: Complete the square
1. Group the \(x\) and \(y\) terms:
\[ (x^2 + 2x) + (y^2 + 4y) = 4 \]
2. Complete the square for \(x\):
\[ x^2 + 2x = (x + 1)^2 - 1 \]
3. Complete the square for \(y\):
\[ y^2 + 4y = (y + 2)^2 - 4 \]
4. Substitute back into the equation:
\[ (x + 1)^2 - 1 + (y + 2)^2 - 4 = 4 \]
\[ (x + 1)^2 + (y + 2)^2 - 5 = 4 \]
\[ (x + 1)^2 + (y + 2)^2 = 9 \]

This shows that the center of the circle is at \((-1, -2)\) and the radius is \(3\).

### Step 3: Find the radius of the circles passing through the point \((-1, 4)\)
Let the center of the circles be \(C(h, k)\). The distance from the center \(C\) to the point \((-1, 4)\) gives the radius \(r\):
\[ r = \sqrt{(h + 1)^2 + (k - 4)^2} \]

### Step 4: Express the distance in terms of \(h\) and \(k\)
The distance \(r\) can be expressed as:
\[ r = \sqrt{(h + 1)^2 + (k - 4)^2} \]

### Step 5: Substitute the center coordinates
Since the center \((h, k)\) lies on the circle \((x + 1)^2 + (y + 2)^2 = 9\), we can express \(k\) in terms of \(h\):
\[ k = -2 \pm \sqrt{9 - (h + 1)^2} \]

### Step 6: Find the maximum and minimum radii
1. The maximum radius \(r_1\) occurs when \(k\) is at its minimum value (i.e., \(k = -2 - \sqrt{9 - (h + 1)^2}\)).
2. The minimum radius \(r_2\) occurs when \(k\) is at its maximum value (i.e., \(k = -2 + \sqrt{9 - (h + 1)^2}\)).

### Step 7: Calculate \(r_1\) and \(r_2\)
1. For maximum radius \(r_1\):
\[ r_1 = \sqrt{(h + 1)^2 + \left(-2 - \sqrt{9 - (h + 1)^2} - 4\right)^2} \]
Simplifying gives:
\[ r_1 = \sqrt{(h + 1)^2 + (-6 - \sqrt{9 - (h + 1)^2})^2} \]

2. For minimum radius \(r_2\):
\[ r_2 = \sqrt{(h + 1)^2 + \left(-2 + \sqrt{9 - (h + 1)^2} - 4\right)^2} \]
Simplifying gives:
\[ r_2 = \sqrt{(h + 1)^2 + (-6 + \sqrt{9 - (h + 1)^2})^2} \]

### Step 8: Find the ratio \( \frac{r_1}{r_2} \)
Given that \( \frac{r_1}{r_2} = a + b\sqrt{2} \), we can find the values of \(a\) and \(b\).

### Step 9: Solve for \(a + b\)
From the calculations, we find:
\[ \frac{r_1}{r_2} = 3 + 0\sqrt{2} \]
Thus, \(a = 3\) and \(b = 0\).

### Final Answer
The value of \(a + b\) is:
\[ a + b = 3 + 0 = 3 \]