Introduction:
To find the focus of a parabola when the equation of the tangent at the vertex and two tangents are known, we can make use of the properties of a parabola and the concept of the directrix. The directrix is a line perpendicular to the axis of symmetry and is equidistant from the focus and vertex of the parabola. By finding the equation of the directrix, we can determine the focus of the parabola.

Steps:

Step 1: Identify the vertex:
The vertex is the point where the parabola intersects the axis of symmetry. It can be obtained by finding the midpoint between the two given tangents.

Step 2: Determine the slope of the tangent at the vertex:
Since the tangent at the vertex is known, we can find its slope using the equation of the tangent line.

Step 3: Find the equation of the directrix:
The slope of the directrix is the negative reciprocal of the slope of the tangent at the vertex. Using this slope and the vertex coordinates, we can find the equation of the directrix.

Step 4: Find the distance between the vertex and the directrix:
The distance between the vertex and the directrix is equal to the distance between the vertex and the focus of the parabola.

Step 5: Calculate the coordinates of the focus:
Using the distance between the vertex and the directrix, we can determine the coordinates of the focus. Since the directrix is perpendicular to the axis of symmetry, the x-coordinate of the focus is the same as the x-coordinate of the vertex. The y-coordinate of the focus can be found by subtracting the distance between the vertex and the directrix from the y-coordinate of the vertex.

Step 6: Write the final answer:
The coordinates of the focus represent the focus of the parabola.

Conclusion:
By following these steps, we can find the focus of a parabola when the equation of the tangent at the vertex and two tangents are known. It is important to understand the properties of a parabola, such as the directrix and the concept of the vertex, to solve such problems effectively.