To solve the problem, we need to find the polars of the point (1, -2) with respect to the two given circles and show that they coincide. Then, we will find another point whose polars with respect to these circles are the same.
### Step 1: Write the equations of the circles
The equations of the circles are:
1. Circle 1: \( x^2 + y^2 + 6y + 5 = 0 \)
2. Circle 2: \( x^2 + y^2 + 2x + 8y + 5 = 0 \)
### Step 2: Convert the equations to standard form
For Circle 1:
\[
x^2 + y^2 + 6y + 5 = 0 \implies x^2 + (y + 3)^2 = 4
\]
This is a circle centered at (0, -3) with radius 2.
For Circle 2:
\[
x^2 + y^2 + 2x + 8y + 5 = 0 \implies (x + 1)^2 + (y + 4)^2 = 2
\]
This is a circle centered at (-1, -4) with radius \(\sqrt{2}\).
### Step 3: Find the polar of the point (1, -2) with respect to Circle 1
The polar of a point \((x_1, y_1)\) with respect to a circle given by \(x^2 + y^2 + 2gx + 2fy + c = 0\) is given by:
\[
xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0
\]
For Circle 1, \(g = 0\), \(f = 3\), and \(c = 5\). Thus, substituting \(x_1 = 1\) and \(y_1 = -2\):
\[
x(1) + y(-2) + 0(x + 1) + 3(y - 2) + 5 = 0
\]
This simplifies to:
\[
x - 2y + 3y - 6 + 5 = 0 \implies x + y - 1 = 0
\]
So, the polar \(L_1\) for Circle 1 is:
\[
x + y - 1 = 0
\]
### Step 4: Find the polar of the point (1, -2) with respect to Circle 2
For Circle 2, \(g = 1\), \(f = 4\), and \(c = 5\). Thus, substituting \(x_1 = 1\) and \(y_1 = -2\):
\[
x(1) + y(-2) + 1(x + 1) + 4(y - 2) + 5 = 0
\]
This simplifies to:
\[
x - 2y + x + 1 + 4y - 8 + 5 = 0 \implies 2x + 2y - 2 = 0 \implies x + y - 1 = 0
\]
So, the polar \(L_2\) for Circle 2 is:
\[
x + y - 1 = 0
\]
### Step 5: Conclusion for the first part
Since both polars \(L_1\) and \(L_2\) are the same, we have proved that the polars of the point (1, -2) with respect to both circles coincide.
### Step 6: Find another point whose polars coincide
Let the new point be \((\alpha, \beta)\). The polars with respect to both circles must also yield the same line.
From Circle 1:
\[
\alpha x + \beta y + 3(y + \beta/2) + 5 = 0
\]
From Circle 2:
\[
\alpha x + \beta y + 2(x + \alpha/2) + 8(y + \beta/2) + 5 = 0
\]
Setting the two equations equal gives us a system of equations to solve for \(\alpha\) and \(\beta\).
### Step 7: Solve the equations
1. From the first polar:
\[
\alpha + 3\beta + 5 = 0 \quad \text{(1)}
\]
2. From the second polar:
\[
2\alpha + 4\beta + 5 = 0 \quad \text{(2)}
\]
From (1), we can express \(\alpha\) in terms of \(\beta\):
\[
\alpha = -3\beta - 5
\]
Substituting into (2):
\[
2(-3\beta - 5) + 4\beta + 5 = 0
\]
\[
-6\beta - 10 + 4\beta + 5 = 0 \implies -2\beta - 5 = 0 \implies \beta = -\frac{5}{2}
\]
Substituting back to find \(\alpha\):
\[
\alpha = -3(-\frac{5}{2}) - 5 = \frac{15}{2} - 5 = \frac{5}{2}
\]
### Final Answer
The coordinates of the other point whose polars with respect to both circles are the same is:
\[
\left(\frac{5}{2}, -\frac{5}{2}\right)
\]
