If `theta _(1), theta_(2)` are the angles of inclination of
tangents through a point P to the circle
`x^(2)+y^(2) = a^(2) ` then find the locus of P when
`cot theta_(1) + cot theta_(2) = k.`

Answer

To solve the problem, we need to find the locus of the point \( P(h, k) \) from which two tangents are drawn to the circle given by the equation \( x^2 + y^2 = a^2 \), under the condition that \( \cot \theta_1 + \cot \theta_2 = k \).

### Step-by-step Solution:

1. **Understand the Geometry**:
The circle has its center at the origin \( O(0, 0) \) and a radius \( a \). The point \( P(h, k) \) lies outside the circle, and from this point, two tangents can be drawn to the circle.

2. **Equation of the Tangents**:
The slopes of the tangents from point \( P(h, k) \) to the circle can be represented as \( m_1 \) and \( m_2 \). The angles of inclination of these tangents are \( \theta_1 \) and \( \theta_2 \), where \( m_1 = \tan \theta_1 \) and \( m_2 = \tan \theta_2 \).

3. **Using the Condition**:
The given condition is \( \cot \theta_1 + \cot \theta_2 = k \). We can rewrite this as:
\[ \frac{1}{\tan \theta_1} + \frac{1}{\tan \theta_2} = k \implies \frac{m_1 + m_2}{m_1 m_2} = k \]
This implies:
\[ m_1 + m_2 = k m_1 m_2 \]

4. **Quadratic Equation**:
The slopes \( m_1 \) and \( m_2 \) are the roots of the quadratic equation derived from the geometry of the tangents:
\[ m^2 - (m_1 + m_2)m + m_1 m_2 = 0 \]
Substituting \( m_1 + m_2 = k m_1 m_2 \) into this equation gives:
\[ m^2 - (k m_1 m_2)m + m_1 m_2 = 0 \]

5. **Finding the Relationship**:
The condition for tangents from point \( P(h, k) \) to the circle can be expressed as:
\[ k^2 - 2h = a^2 \]
Rearranging gives:
\[ 2h = k^2 - a^2 \implies h = \frac{k^2 - a^2}{2} \]

6. **Finding the Locus**:
The locus of point \( P \) can be expressed in terms of \( k \):
\[ h = \frac{k^2 - a^2}{2} \]
This shows that as \( k \) varies, \( h \) will vary accordingly. The value of \( k \) can take any real number, and thus, the locus will be a parabola.

7. **Final Locus Equation**:
The locus of point \( P(h, k) \) can be expressed as:
\[ h = \frac{1}{2}(k^2 - a^2) \]
This is the equation of a parabola that opens to the right.