Locus of the point from which the difference of the squares of lengths of tangents drawn to two given circles is constant is

Options

A. circle
B. parabola
C. straight line
D. none

Answer

To find the locus of the point from which the difference of the squares of the lengths of tangents drawn to two given circles is constant, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the locus of points from which the difference of the squares of the lengths of tangents drawn to two circles is constant. Let the two circles be represented by their equations.

2. **Equations of the Circles**:
Let the equations of the two circles be:
- Circle 1: \( x^2 + y^2 + 2G_1x + 2F_1y + C_1 = 0 \)
- Circle 2: \( x^2 + y^2 + 2G_2x + 2F_2y + C_2 = 0 \)

3. **Length of Tangents**:
The length of the tangent from a point \( (A, B) \) to a circle can be given by the formula:
- For Circle 1:
\[ L_1 = \sqrt{A^2 + B^2 + 2G_1A + 2F_1B + C_1} \]
- For Circle 2:
\[ L_2 = \sqrt{A^2 + B^2 + 2G_2A + 2F_2B + C_2} \]

4. **Setting Up the Equation**:
According to the problem, the difference of the squares of the lengths of the tangents is constant:
\[ L_1^2 - L_2^2 = K \]
where \( K \) is a constant.

5. **Substituting the Lengths**:
Substitute \( L_1^2 \) and \( L_2^2 \):
\[ (A^2 + B^2 + 2G_1A + 2F_1B + C_1) - (A^2 + B^2 + 2G_2A + 2F_2B + C_2) = K \]

6. **Simplifying the Equation**:
Simplifying the above equation gives:
\[ (2G_1 - 2G_2)A + (2F_1 - 2F_2)B + (C_1 - C_2 - K) = 0 \]

7. **Rearranging the Terms**:
Rearranging the terms, we can express the equation in the form:
\[ (G_1 - G_2)A + (F_1 - F_2)B = \frac{K + C_2 - C_1}{2} \]

8. **Identifying the Locus**:
The equation derived is a linear equation in \( A \) and \( B \), which represents a straight line. Thus, the locus of the point from which the difference of the squares of the lengths of tangents drawn to the two circles is constant is a straight line.

### Final Answer:
The locus of the point from which the difference of the squares of lengths of tangents drawn to two given circles is constant is a **straight line**.