Find the locus of a point whose distance from a fixed point is in a constant ratio to the tangent drawn from it to a given circle.

Answer

To find the locus of a point whose distance from a fixed point is in a constant ratio to the tangent drawn from it to a given circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( P(h, k) \) be the variable point whose locus we need to find.
- Let \( F(\alpha, \beta) \) be the fixed point.
- Let the given circle be defined by the equation \( x^2 + y^2 = a^2 \).

2. **Distance from Fixed Point**:
- The distance \( PQ \) from the variable point \( P(h, k) \) to the fixed point \( F(\alpha, \beta) \) is given by:
\[ PQ = \sqrt{(h - \alpha)^2 + (k - \beta)^2} \]

3. **Length of Tangent**:
- The length of the tangent \( PT \) from point \( P(h, k) \) to the circle \( x^2 + y^2 = a^2 \) is given by:
\[ PT = \sqrt{h^2 + k^2 - a^2} \]

4. **Set Up the Ratio**:
- According to the problem, the distance \( PQ \) is in a constant ratio \( \lambda \) to the length of the tangent \( PT \):
\[ PQ = \lambda PT \]

5. **Square Both Sides**:
- Squaring both sides gives:
\[ PQ^2 = \lambda^2 PT^2 \]

6. **Substituting the Distances**:
- Substitute the expressions for \( PQ^2 \) and \( PT^2 \):
\[ (h - \alpha)^2 + (k - \beta)^2 = \lambda^2 (h^2 + k^2 - a^2) \]

7. **Expand Both Sides**:
- Expanding the left side:
\[ h^2 - 2h\alpha + \alpha^2 + k^2 - 2k\beta + \beta^2 \]
- The right side becomes:
\[ \lambda^2 h^2 + \lambda^2 k^2 - \lambda^2 a^2 \]

8. **Rearranging the Equation**:
- Rearranging the equation gives:
\[ (1 - \lambda^2)h^2 + (1 - \lambda^2)k^2 - 2h\alpha - 2k\beta + (\alpha^2 + \beta^2 + \lambda^2 a^2) = 0 \]

9. **Identifying the Locus**:
- This is a standard form of a conic section. If \( 1 - \lambda^2 \neq 0 \), we can divide through by \( 1 - \lambda^2 \) to get:
\[ h^2 + k^2 - \frac{2h\alpha}{1 - \lambda^2} - \frac{2k\beta}{1 - \lambda^2} + \frac{\alpha^2 + \beta^2 + \lambda^2 a^2}{1 - \lambda^2} = 0 \]
- Replacing \( h \) and \( k \) with \( x \) and \( y \), we have the equation of a circle or a parabola depending on the value of \( \lambda \).

### Final Locus Equation:
The locus of the point \( P \) is a conic section, which can be specifically described as a circle or parabola depending on the constant ratio \( \lambda \).