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Solution

Let the coordinates of $P$ be $(x_1, y_1)$. Equation of chord of contact to $y^2 = 4a x$ is $T_1 = 0 \Rightarrow y y_1 = 2a(x + x_1).$ This line touches $x^2 = 4b y$. Substitute $y = \dfrac{x^2}{4b}$ in the line equation: $\dfrac{x^2 y_1}{4b} = 2a(x + x_1)$ $\Rightarrow y_1 x^2 - 8abx - 8abx_1 = 0.$ For tangency, discriminant $= 0$: $(8ab)^2 - 4y_1(-8abx_1) = 0$ $\Rightarrow 64a^2b^2 + 32abx_1 y_1 = 0$ $\Rightarrow 2x_1 y_1 + 4ab = 0 \Rightarrow x_1 y_1 = -2ab.$ Thus, locus of $P$ is $x y = -2ab$, which represents a **hyperbola**.