Subtangent and Subnormal - Practice Questions & MCQ

Length of Tangent, Subtangent, Normal and Subnormal

Let $P(x, y)$ is any point on the parabola $y^2=4 a x$. The tangent and normal at point $P(x, y)$ meets axis of the parabola at $T$ and $G$ respectively and let tangent at point $P(x, y)$ makes an angle ? with $X$ axis.

A

Then, we define
PT = Length of Tangent
PG = Length of Normal
TN = Length of Subtangent
And, NG = length of Subnormal

$
\begin{array}{ll}
\because & \mathrm{PN}=y \\
\therefore & \mathrm{PT}=\mathrm{PN} \csc \psi=y \csc \psi \\
& \mathrm{PG}=\mathrm{PN} \csc \left(90^{\circ}-\psi\right)=y \sec \psi \\
& \text { TN }=\text { PN } \cot (\psi)=y \cot \psi \\
\text { and } & \mathrm{NG}=\mathrm{PN} \cot \left(90^{\circ}-\psi\right)=y \tan \psi
\end{array}
$

where, $\tan \psi=\frac{2 a}{y}=m \quad[$ slope of tangent at P$]$