Question

Find the equation of chord joining the two parametric points of the $P\left(t_1\right)\&P\left(t_2\right)$ of the hyperbola $xy=c^2$.

• A. \)
• B. \)
• C. \)
• D. \)

Answer 1

The correct option is D

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Equation of hyperbola is $xy=c^2$

parametric point on oblique hyperbola is

$x=ct,y=\frac{c}{t}$ where $tϵR-\left\{0\right\}$

$p\left(t_1\right)=(c{t}_1,\frac{c}{t_1})$

$p\left(t_2\right)=(c{t}_2,\frac{c}{t_2})$

Equation of the chord is

$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

$y-\frac{c}{t_1}=\frac{\frac{c}{t_2}-\frac{c}{t_1}}{c(t_2-t_1)}(x-c{t}_1)$

$\frac{t_{1}y-c}{t_1}=\frac{c(t_1-t_2)}{t_1.t_{2}c(t_2-t_1)}(x-c{t}_1)$

$\frac{t_{1}y-c}{t_1}=\frac{-1}{t_1{t}_2}(x-c{t}_1)$

$t_1{t}_{2}y-t_{2}c=-x+c{t}_1$

$x+t_1{t}_{2}y=c{t}_1+c{t}_2$

$x+t_1{t}_{2}y=c(t_1+t_2)$