JEE Main & Advanced Mathematics Conic Sections Length of Tangent, Subtangent, Normal, Subnormal

Let the parabola \[{{y}^{2}}=4ax\]. Let the tangent and normal at \[P({{x}_{1}},{{y}_{1}})\] meet the axis of parabola at T and G respectively, and tangent at \[P({{x}_{1}},{{y}_{1}})\] makes angle \[\psi \] with the positive direction of x-axis. \[A(0,\,0)\] is the vertex of the parabola and \[PN=y\]. Then,  

(1) Length of tangent \[=PT=PN\,\text{cosec}\,\,\psi ={{y}_{1}}\,\text{cosec}\,\psi \]   (2) Length of normal \[=PG=PN\text{cosec}({{90}^{o}}-\psi )={{y}_{1}}\sec \psi \]   (3) Length of subtangent \[=TN=PN\cot \psi ={{y}_{1}}\cot \psi \]   (4) Length of subnormal \[=NG=PN\cot ({{90}^{o}}-\psi )={{y}_{1}}\tan \psi \]   where, \[\tan \psi =\frac{2a}{{{y}_{1}}}=m\],  [Slope of tangent at \[P(x,\,\,y)\]]