Hyperbola
9.0 Chord bisected at a given point
9.0 Chord bisected at a given point
The equation of chord of hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ whose middle point is $({x_1},{y_1})$ can be find out using $$T = {S_1}$$
where $$T \equiv \frac{{x{x_1}}}{{{a^2}}} - \frac{{y{y_1}}}{{{b^2}}} - 1;{\text{ }}{S_1} \equiv \frac{{{x_1}^2}}{{{a^2}}} - \frac{{{y_1}^2}}{{{b^2}}} - 1$$
Question 8. Find the locus of mid-point of focal chords of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$.
Solution: Let $P(h,k)$ be the mid-point of focal chord and the equation of chord whose mid-point is given can be find out by $$\frac{{xh}}{{{a^2}}} - \frac{{yk}}{{{b^2}}} - 1 = \frac{{{h^2}}}{{{a^2}}} - \frac{{{k^2}}}{{{b^2}}} - 1$$
Since, it is a focal chord, it passes through focus, either $(ae,0)$ or $(-ae,0)$.
If it passes through $(ae,0)$, locus is $$\frac{{ex}}{a} = \frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}}$$
If it passes through $(-ae,0)$, locus is $$ - \frac{{ex}}{a} = \frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}}$$
