Hyperbola
1.0 Definition
2.0 Standard Equation of Hyperbola
3.0 Difference between two forms of Hyperbola
4.0 Parametric Co-ordinates
5.0 Equation of tangent to Hyperbola
6.0 Equation of normal to Hyperbola
7.0 Pair of tangents
8.0 Chord of contact
9.0 Chord bisected at a given point
10.0 Asymptotes
11.0 Rectangular Hyperbola
5.2 Parametric form
If the coordinates of a point on which tangent is drawn is in the form of parametric coordinates i.e., $(a\sec \theta ,b\tan \theta )$ then equation of tangent to the hyperbola is $$\frac{{x\sec \theta }}{a} - \frac{{y\tan \theta }}{b} = 1$$
Question 4. Find the equation and the length of common tangents to hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ and $\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1$.
Solution: The equation of tangent to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ drawn at a point $(a\sec \phi ,b\tan \phi )$ is $$\frac{x}{a}\sec \phi - \frac{y}{b}\tan \phi = 1...(1)$$ Similarly, the equation of tangent to the hyperbola $\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1$ drawn at a point $(b\tan \theta ,a\sec \theta )$ is $$\frac{y}{a}\sec \theta - \frac{x}{b}\tan \theta = 1...(2)$$ If equations $(1)$ and $(2)$ are common tangents then they should be identical. On comparing the co-efficients of $x$ and $y$, we get $$\frac{{\sec \theta }}{a} = - \frac{{\tan \phi }}{b}...(3)$$ and $$ - \frac{{\tan \theta }}{b} = \frac{{\sec \phi }}{a}{\text{ or }}\sec \theta = - \frac{a}{b}\tan \phi ...(4)$$
$\because {\sec ^2}\theta - {\tan ^2}\theta = 1$, from equation $(3)$ and $(4)$,
$$\frac{{{a^2}}}{{{b^2}}}{\tan ^2}\phi - \frac{{{b^2}}}{{{a^2}}}{\sec ^2}\phi = 1$$$$\frac{{{a^2}}}{{{b^2}}}{\tan ^2}\phi - \frac{{{b^2}}}{{{a^2}}}(1 + {\tan ^2}\phi ) = 1$$$$(\frac{{{a^2}}}{{{b^2}}} - \frac{{{b^2}}}{{{a^2}}}){\tan ^2}\phi = 1 + \frac{{{b^2}}}{{{a^2}}}$$$${\tan ^2}\phi = \frac{{{b^2}}}{{{a^2} - {b^2}}}$$ and $${\sec ^2}\phi = 1 + {\tan ^2}\phi = \frac{{{a^2}}}{{{a^2} - {b^2}}}$$
Hence, point of contact are $$\{ \pm \frac{{{a^2}}}{{\sqrt {{a^2} - {b^2}} }}, \pm \frac{{{b^2}}}{{\sqrt {{a^2} - {b^2}} }}\} {\text{ and }}\{ \pm \frac{{{b^2}}}{{\sqrt {{a^2} - {b^2}} }}, \pm \frac{{{a^2}}}{{\sqrt {{a^2} - {b^2}} }}\} $$
Length of common tangent i.e., the distance between above two points using distance formulae is $$\sqrt 2 \frac{{{a^2} + {b^2}}}{{\sqrt {{a^2} - {b^2}} }}$$ and the equation of common tangent by putting the values of $\sec \phi $ and $\tan \phi $ in equation $(1)$ is $$x \mp y = \pm \sqrt {{a^2} - {b^2}} $$
