A tangent to the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ meets the x-axis in T and the tangent at the vertex A in P. If TAPQ is a rectangle, then the locus of Q is

$$\begin{gathered} \mathrm{Given} \mathrm{parabola} {\mathrm{y}}^2=4\mathrm{ax} \\ Let Q be (h,k),T(-h,0) and P(0,k) \\ Let the \tan gent be, y=mx+\frac{a}{m} \\ \Rightarrow \frac{a}{m}=-hm \\ \Rightarrow m^2=\frac{-a}{h}...\left(1\right) \\ \frac{a}{m}=k\Rightarrow m=\frac{a}{k}...\left(2\right) \\ From \left(1\right) and \left(2\right) \\ \frac{a^2}{k^2}=\frac{-a}{h} \\ \Rightarrow ah=-k^2 \\ locus of Q(h,k) is \\ \Rightarrow ax=-y^2 \\ \Rightarrow y^2+ax=0 \end{gathered}$$