If on a given base BC[B(0,0) and C(2,0)], a triangle is described such that the sum of the base angles is 4, then the equation of the locus of the opposite vertex A is parabola whose directrix is y=k. The value of k is _________ .

Explanation

To solve the problem, we need to find the locus of the vertex A of triangle ABC, where B(0,0) and C(2,0) are the base points, and the sum of the base angles at B and C is 4.

### Step-by-Step Solution:

1. **Identify the Points and Angles**:
- Let point A have coordinates (x, y).
- The angles at points B and C are denoted as α (at B) and β (at C).
- The sum of the angles α + β = 4.

2. **Using the Tangent Function**:
- From triangle ADB, we have:
\[ \tan(\alpha) = \frac{y}{x} \]
- From triangle ADC, we have:
\[ \tan(\beta) = \frac{y}{2 - x} \]

3. **Setting Up the Equation**:
- Given that the sum of the tangents of the angles is 4:
\[ \tan(\alpha) + \tan(\beta) = 4 \]
- Substituting the expressions for tan(α) and tan(β):
\[ \frac{y}{x} + \frac{y}{2 - x} = 4 \]

4. **Finding a Common Denominator**:
- The common denominator for the left-hand side is \(x(2 - x)\):
\[ \frac{y(2 - x) + yx}{x(2 - x)} = 4 \]
- This simplifies to:
\[ \frac{y(2)}{x(2 - x)} = 4 \]

5. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ 2y = 4x(2 - x) \]
- Simplifying this, we have:
\[ 2y = 8x - 4x^2 \]
- Dividing by 2:
\[ y = 4x - 2x^2 \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ 2x^2 - 4x + y = 0 \]

7. **Identifying the Parabola**:
- This can be rewritten in standard form:
\[ x^2 - 2x + \frac{y}{2} = 0 \]
- Completing the square for x:
\[ (x - 1)^2 = \frac{y}{2} + 1 \]

8. **Finding the Directrix**:
- The standard form of a parabola is:
\[ (x - h)^2 = -4a(y - k) \]
- From our equation, we can identify:
- Vertex (h, k) = (1, 2)
- The coefficient of y gives us \(4a = \frac{1}{2}\), thus \(a = \frac{1}{8}\).
- The directrix is given by:
\[ y = k - a = 2 - \frac{1}{8} = \frac{16}{8} - \frac{1}{8} = \frac{15}{8} \]

9. **Final Answer**:
- The value of k, which is the y-coordinate of the directrix, is:
\[ k = \frac{15}{8} \]