Question:

Step 1: Parametrize the points on the parabola. Let us take two points on the parabola $y^2 = 4ax$ in parametric form: $$P = (at_1^2, 2at_1) \quad \text{and} \quad Q = (at_2^2, 2at_2)$$ Step 2: Write the equations of tangents at P and Q. The equation of the tangent to the parabola $y^2 = 4ax$ at a point $(at^2, 2at)$ is given by $ty = x + at^2$. Therefore: - Tangent at P: $t_1 y = x + at_1^2$ - Tangent at Q: $t_2 y = x + at_2^2$ Step 3: Find the condition for tangents to meet on the directrix. The directrix of the parabola $y^2 = 4ax$ is $x = -a$. For the tangent at P to pass through the directrix, substitute $x = -a$: $$t_1 y = -a + at_1^2 = a(t_1^2 - 1)$$ $$y = \frac{a(t_1^2 - 1)}{t_1}$$ Similarly, for the tangent at Q: $$y = \frac{a(t_2^2 - 1)}{t_2}$$ Step 4: Apply the condition that both tangents meet at the same point R on the directrix. Since both tangents meet at point R on the directrix, the y-coordinates must be equal: $$\frac{a(t_1^2 - 1)}{t_1} = \frac{a(t_2^2 - 1)}{t_2}$$ Simplifying: $$t_2(t_1^2 - 1) = t_1(t_2^2 - 1)$$ $$t_1^2 t_2 - t_2 = t_1 t_2^2 - t_1$$ $$t_1^2 t_2 - t_1 t_2^2 = t_2 - t_1$$ $$t_1 t_2(t_1 - t_2) = -(t_1 - t_2)$$ Since $t_1 \neq t_2$, we can divide by $(t_1 - t_2)$: $$t_1 t_2 = -1$$ Step 5: Identify the geometric property of PQ. The condition $t_1 t_2 = -1$ is the characteristic property of a focal chord of the parabola. This means that the chord PQ passes through the focus $F = (a, 0)$. Step 6: Determine the locus of the circumcenter. For a focal chord PQ of a parabola, the circumcenter of triangle PQR (where R is the point where the tangents at P and Q meet on the directrix) traces a specific locus. This locus is itself a parabola. Step 7: Find the focus of the locus parabola. Through the geometric properties of focal chords and their associated circumcenters in a parabola, the locus of the circumcenter of triangle PQR is a parabola whose focus is located at: $$\boxed{\left(\frac{3a}{2}, 0\right)}$$ **Answer: Option 3** — The focus of the locus of the circumcenter of $\triangle PQR$ is $\left(\frac{3a}{2}, 0\right)$.