Find the locus of the intersection of tangents
which meet at a given angle `alpha `

Answer

To find the locus of the intersection of tangents that meet at a given angle \(\alpha\) to an ellipse, we will follow these steps:

### Step 1: Understand the Equation of the Ellipse
The standard equation of an ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

### Step 2: Write the Equation of the Tangents
The equation of the tangents to the ellipse at a point \((x_1, y_1)\) is given by:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
However, since we are interested in the intersection of two tangents, we can express the tangents in terms of the slope \(m\):
\[ y = mx + \frac{b^2}{a^2} \cdot \frac{1}{m} \]

### Step 3: Determine the Condition for the Angle Between Tangents
For two tangents to meet at an angle \(\alpha\), we can use the formula for the angle between two lines:
\[ \tan(\alpha) = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| \]
where \(m_1\) and \(m_2\) are the slopes of the tangents.

### Step 4: Set Up the Quadratic Equation
From the tangents, we can derive a quadratic equation in terms of \(m\):
\[ \beta^2 + m^2 \alpha^2 - 2m \alpha \beta = a^2 m^2 + b^2 \]
Rearranging gives:
\[ (m^2 (a^2 + \alpha^2) - 2m \alpha \beta + \beta^2 - b^2 = 0 \]

### Step 5: Solve for \(m_1\) and \(m_2\)
Using the quadratic formula, we can find \(m_1\) and \(m_2\):
\[ m = \frac{2\alpha \beta \pm \sqrt{(2\alpha \beta)^2 - 4(a^2 + \alpha^2)(\beta^2 - b^2)}}{2(a^2 + \alpha^2)} \]

### Step 6: Find the Locus of the Intersection Point
To find the locus of the intersection point \(P\) of the tangents, we need to express the coordinates \(x\) and \(y\) in terms of \(m\) and \(\alpha\):
\[ \tan(\alpha) = \frac{m_1 - m_2}{1 + m_1 m_2} \]
Substituting the values of \(m_1\) and \(m_2\) leads to a relationship involving \(x\) and \(y\).

### Step 7: Simplify and Rearrange
After some algebraic manipulation, we arrive at the final equation representing the locus of point \(P\):
\[ x^2 b^2 + y^2 a^2 - a^2 b^2 = k^2 \]
where \(k\) is a constant derived from \(\tan(\alpha)\).

### Conclusion
The locus of the intersection of the tangents that meet at an angle \(\alpha\) is an ellipse.

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