Length of Tangent and Normal and Subtangent and subnormal - Practice Questions & MCQ

Length of Tangent, Normal, Subtangent and Subnormal  

Length of Tangent:

The length of the portion lying between the point of tangency i.e. the point on the curve from which a tangent is drawn and the point where the tangent meets the x-axis. Here the point of tangency is $\mathrm{P}\left(\mathrm{x}_0\right.$, $\left.\mathrm{y}_0\right)$
In the figure, the length of segment PT is the length of the tangent.
In $\triangle \mathrm{PTS}$

$
\begin{aligned}
\mathrm{PT} & =|y \cdot \csc \theta|=|y| \sqrt{1+\cot ^2 \theta} \\
& =|\mathrm{y}| \sqrt{1+\left(\frac{\mathrm{dx}}{\mathrm{dy}}\right)_{\left(\mathrm{x} 0, \mathrm{y}_0\right)}}
\end{aligned}
$
Length of Normal:
A segment of normal PN is called the length of Normal.
In $\triangle \mathrm{PSN}$

$
\begin{aligned}
\mathrm{PN} & =\left|y \cdot \csc \left(90^{\circ}-\theta\right)\right|=|y \cdot \sec \theta| \\
& =|\mathrm{y}| \sqrt{1+\tan ^2 \theta}=|\mathrm{y}| \sqrt{1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\left(\mathrm{x}_0, \mathrm{y}_0\right)}}
\end{aligned}
$

Length of Subtangent:

The projection of the segment PT along the x-axis is called the length of the subtangent. In the figure, ST is the length of the subtangent.
In $\triangle \mathrm{PST}$

$
\begin{aligned}
\mathrm{ST} & =|y \cdot \cot \theta|=\left|\frac{y}{\tan \theta}\right| \\
& =\left|\mathrm{y} \cdot \frac{\mathrm{dx}}{\mathrm{dy}}\right|
\end{aligned}
$
Length of Subnormal:
The projection of the segment PN along the x-axis is called the length of the subnormal. In the figure, SN is the length of subnormal.
In $\triangle \mathrm{PSN}$

$
\begin{aligned}
\mathrm{SN} & =\left|y \cdot \cot \left(90^{\circ}-\theta\right)\right|=|y \cdot \tan \theta| \\
& =\left|\mathrm{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}\right|
\end{aligned}
$