The length of the chord of the parabola ${\mathrm{x}}^2=4\mathrm{ay}$ passing through the vertex and having slope $\tan \mathrm{\alpha }$ is

$$\begin{gathered} \mathrm{Given} \mathrm{parabola} {\mathrm{x}}^2=4\mathrm{ay} \\ vertex\left(O\right)=(0,0) \\ Let A=(a{t}^2,2at) be any point on parabola \\ now slope of OA=\frac{{\mathrm{at}}^2}{2\mathrm{at}} \\ \tan \mathrm{\alpha }=\frac{{\mathrm{at}}^2}{2\mathrm{at}}=\frac{\mathrm{t}}{2} \\ \Rightarrow \mathrm{t}=2 \tan \mathrm{\alpha } \\ Now \mathrm{OA}=\sqrt{4{\mathrm{a}}^2{\mathrm{t}}^2+{\mathrm{a}}^2{\mathrm{t}}^4}=\mathrm{at}\sqrt{4+{\mathrm{t}}^2} \\ =2\mathrm{a} \tan \mathrm{\alpha }(2 \sec \mathrm{\alpha }) \\ =4\mathrm{a} \mathrm{sec\alpha } \tan \mathrm{\alpha } \end{gathered}$$