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
11.2 Important results
- Equation of tangent at a point $P({x_1},{y_1})$ to the rectangular hyperbola $xy = {c^2}$ is $$\frac{x}{{{x_1}}} + \frac{y}{{{y_1}}} = 2$$
Proof: As we know that the equation of tangent at a point $P({x_1},{y_1})$ to the rectangular hyperbola $xy = {c^2}$ can be find out using $T=0$. Therefore, $$\frac{{x{y_1}}}{2} + \frac{{y{x_1}}}{2} = {c^2}$$ Divide by ${x_1}{y_1}$ we get $$\frac{x}{{{x_1}}} + \frac{y}{{{y_1}}} = 2\frac{{{c^2}}}{{{x_1}{y_1}}}$$ Since the point $P({x_1},{y_1})$ lies on the hyperbola, it must satisfy the equation of hyperbola i.e., $${x_1}{y_1} = {c^2}$$ Put the value of ${c^2}$, we get $$\frac{x}{{{x_1}}} + \frac{y}{{{y_1}}} = 2$$
- Equation of tangent at a point $P(t)$ to the rectangular hyperbola $xy = {c^2}$ is $$\frac{x}{t} + ty = 2c$$
- Point of intersection of tangents at ${t_1}$ and ${t_2}$ to the rectangular hyperbola $xy = {c^2}$ is $$(\frac{{2c{t_1}{t_2}}}{{{t_1} + {t_2}}},\frac{{2c}}{{{t_1} + {t_2}}})$$
- Equation of chord joining the points $P({t_1})$ and $Q({t_2})$ is $$x + {t_1}{t_2}y = c({t_1} + {t_2})$$
- Equation of normal at a point $P({x_1},{y_1})$ to the rectangular hyperbola $xy = {c^2}$ is $$x{x_1} - y{y_1} = {x_1}^2 - {y_1}^2$$
- Equation of normal at $P(t)$ to the rectangular hyperbola $xy = {c^2}$ is $$x{t^3} - yt - c{t^4} + c = 0$$
- Equation of chord to the rectangular hyperbola $xy = {c^2}$ whose middle point is given say $(h,k)$ is $$kx+hy=2hk$$
- Difference of focal distances is a constant i.e., $$\left| {SP - S'P} \right| = 2a$$
- If a circle and a rectangular hyperbola $xy = {c^2}$ meets in four points having parameters ${t_1}$, ${t_2}$, ${t_3}$ and ${t_4}$, then $${t_1}{t_2}{t_3}{t_4} = 1$$ and the centre of the circle through the points ${t_1}$, ${t_2}$ and ${t_3}$ is $$\{ \frac{c}{2}({t_1} + {t_2} + {t_3} + \frac{1}{{{t_1}{t_2}{t_3}}}),\frac{c}{2}(\frac{1}{{{t_1}}} + \frac{1}{{{t_2}}} + \frac{1}{{{t_3}}} + {t_1}{t_2}{t_3})\} $$
