Browse Questions

# Find the length of sub tangent and sub-normal?(for hyperbola)

$\begin{array}{1 1}(a)\;x_1-\large\frac{a^2}{x_1},\normalsize (e^2-1)x_1\\(b)\;x_1-\large\frac{a}{x_1},\normalsize (e^3-1)x_1\\(c)\;x_1^2+\large\frac{a^2}{x_1},\normalsize (e-1)x_1\\(d)\;x_1+\large\frac{a^2}{x_1},\normalsize (e^2+1)x_1\end{array}$

Let the tangent and normal at $4P(x_1,y_1)$ meet the x-axis at $T$ and $C$
Equation of tangent at $P(x_1,y_1)$
$\large\frac{xx_1}{a^2}-\frac{yy_1}{b^2}$$=1------(1) T lies on x-axis ,put y=0 in (1) \Rightarrow x=CT CT=\large\frac{a^2}{x_1} CN=x_1 Length of subtangent NT=CN-CT \Rightarrow x_1-\large\frac{a^2}{x_1} Equation of normal at P(x_1,y_1) is \large\frac{a^2x}{x_1}+\frac{b^2y}{y_1}$$=a^2+b^2$-----(2)
G lies on x-axis ,y=0 in (2)
$x=CG$
$CG=\large\frac{(a^2+b^2)x_1}{a^2}$
Length of sub-normal NG=CG-CN