The lengths of the tangents from any point on the circle `x^(2)+y^(2)+8x+1=0` to the circles `x^(2)+y^(2)+7x+1=0` and `x^(2)+y^(2)+4x+1=0` are in the ratio

Options

A. `1:2`
B. `1:3`
C. `1:4`
D. `1:sqrt2`

Answer

To find the lengths of the tangents from any point on the circle \( x^2 + y^2 + 8x + 1 = 0 \) to the circles \( x^2 + y^2 + 7x + 1 = 0 \) and \( x^2 + y^2 + 4x + 1 = 0 \), we can follow these steps:

### Step 1: Rewrite the equations of the circles
We start by rewriting the equations of the circles in standard form.

1. **First Circle (C1)**:
\[ x^2 + y^2 + 8x + 1 = 0 \implies (x + 4)^2 + y^2 = 15 \]
This circle has center \((-4, 0)\) and radius \(\sqrt{15}\).

2. **Second Circle (C2)**:
\[ x^2 + y^2 + 7x + 1 = 0 \implies (x + \frac{7}{2})^2 + y^2 = \frac{25}{4} \]
This circle has center \((-3.5, 0)\) and radius \(\frac{5}{2}\).

3. **Third Circle (C3)**:
\[ x^2 + y^2 + 4x + 1 = 0 \implies (x + 2)^2 + y^2 = 3 \]
This circle has center \((-2, 0)\) and radius \(\sqrt{3}\).

### Step 2: Length of the tangent from a point on the first circle
Let \( P(h, k) \) be a point on the first circle. The length of the tangent from point \( P \) to circle \( C1 \) is given by:
\[ L_1 = \sqrt{h^2 + k^2 + 7h + 1} \]

The point \( P(h, k) \) lies on the first circle, so it satisfies:
\[ h^2 + k^2 + 8h + 1 = 0 \implies h^2 + k^2 = -8h - 1 \]

### Step 3: Substitute \( h^2 + k^2 \) in \( L_1 \)
Substituting \( h^2 + k^2 \) in the expression for \( L_1 \):
\[ L_1 = \sqrt{(-8h - 1) + 7h + 1} = \sqrt{-8h + 7h} = \sqrt{-h} \]

### Step 4: Length of the tangent to the second circle
The length of the tangent from point \( P(h, k) \) to circle \( C2 \) is given by:
\[ L_2 = \sqrt{h^2 + k^2 + 4h + 1} \]
Substituting \( h^2 + k^2 \) again:
\[ L_2 = \sqrt{(-8h - 1) + 4h + 1} = \sqrt{-8h + 4h} = \sqrt{-4h} \]

### Step 5: Find the ratio of the lengths of the tangents
Now we find the ratio \( \frac{L_1}{L_2} \):
\[ \frac{L_1}{L_2} = \frac{\sqrt{-h}}{\sqrt{-4h}} = \frac{1}{\sqrt{4}} = \frac{1}{2} \]

Thus, the ratio of the lengths of the tangents from any point on the first circle to the other two circles is:
\[ L_1 : L_2 = 1 : 2 \]

### Final Answer
The correct option is \( 1 : 2 \).