To find the area of the triangle inscribed in the parabola y^2 = 4ax with vertices (x1, y1) and (x2, y2), we can use the formula for finding the area of a triangle given its vertices.

The formula for the area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is:

Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

In this case, we have the vertices (x1, y1) and (x2, y2), but we need to find the third vertex (x3, y3) on the parabola y^2 = 4ax.

Since y^2 = 4ax, we can solve for y:

y = ±sqrt(4ax)

We can set y = sqrt(4ax) to find a point on the parabola.

Let's say the x-coordinate of this point is x3. Substituting this into the equation, we have:

sqrt(4ax3) = y3

Squaring both sides, we get:

4ax3 = y3^2

Solving for x3, we have:

x3 = y3^2 / (4a)

Now we have the three vertices of the triangle: (x1, y1), (x2, y2), and (x3, y3).

Plugging these values into the formula for the area of a triangle, we have:

Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + (y3^2 / (4a))(y1 - y2)|

Simplifying further, we have:

Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + y3(y1 - y2) / (4a)|

So the area of the triangle inscribed in the parabola y^2 = 4ax with vertices (x1, y1) and (x2, y2) is 1/2 times the absolute value of the expression above.