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
6.3 Slope form
The equation of normal to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ in terms of slope $m$ is $$y = mx \mp \frac{{m({a^2} + {b^2})}}{{\sqrt {{a^2} - {b^2}{m^2}} }}$$
Question 6. A normal to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ meets the axes in $M$ and $N$ and lines $MP$ and $NP$ are drawn perpendicular to the axes meeting at $P$. Prove that the locus of $P$ is the hyperbola $${a^2}{x^2} - {b^2}{y^2} = {({a^2} + {b^2})^2}$$
Solution: The equation of normal at the point $Q(a\sec \theta ,b\tan \theta )$ to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ is $$ax\cos \theta + by\cot \theta = {a^2} + {b^2}...(1)$$
Since the normal given by equation $(1)$ meets the $X-$axis in $M(\frac{{{a^2} + {b^2}}}{a}\sec \theta ,0)$ and $Y-$axis in $N(0,\frac{{{a^2} + {b^2}}}{b}\tan \theta )$.
$\therefore $ Equation of line $MP$, passing through $M$ and perpendicular to $X-$axis, is $$x = (\frac{{{a^2} + {b^2}}}{a})\sec \theta {\text{ or }}\sec \theta = \frac{{ax}}{{{a^2} + {b^2}}}...(2)$$
and Equation of line $NP$, passing through $N$ and perpendicular to $Y-$axis, is $$y = (\frac{{{a^2} + {b^2}}}{b})\tan \theta {\text{ or }}\tan \theta = \frac{{bx}}{{{a^2} + {b^2}}}...(3)$$
hus, the locus of point of intersection of $MP$ and $NP$ is obtained by eliminating $\theta $ from equations $(2)$ and $(3)$, $$\frac{{{a^2}{x^2}}}{{{{({a^2} + {b^2})}^2}}} - \frac{{{b^2}{y^2}}}{{{{({a^2} + {b^2})}^2}}} = 1$$$${a^2}{x^2} - {b^2}{y^2} = {({a^2} + {b^2})^2}$$
