Name: Equation of Chord of Hyperbola in Parametric Form
Uploaded: 2019-07-19
Duration: 1 min 30 s
Description: How to find the equation of chord of a hyperbola joining two points whose parametric coordinates are known? Check out this video to find out.

Question

If a rectangular hyperbola have the equation,

x y = c^{2},

prove that the locus of the middle points of the chords of constant length

2 d

is

(x^{2} + y^{2}) (x y - c^{2}) = d^{2} x y .

Accepted Answer

Let, $(h, k)$ be the mid-point of the chord $2 d$ of the given rectangular hyperbola $(x^{2} + y^{2}) (x y - c^{2}) = d^{2} x y .$ .

Let, $A = (c t_{1}, \frac{c}{t_{1}}) & B = (c t_{2}, \frac{c}{t_{2}})$ .

$\Rightarrow h = \frac{c t_{1} + c t_{2}}{2} & K = \frac{\frac{c}{t_{1}} + \frac{c}{t_{2}}}{2}$

$\Rightarrow 2 h = c (t_{1} + t_{2}) & 2 k = c (\frac{1}{t_{1}} + \frac{1}{t_{2}})$

$\Rightarrow \frac{2 h}{c} = (t_{1} + t_{2}) & \frac{2 k}{c} = (\frac{t_{1} + t_{2}}{t_{1} \cdot t_{2}})$

$\Rightarrow \frac{2 h}{c} = (t_{1} + t_{2}) & t_{1} \cdot t_{2} = (\frac{\frac{2 h}{c}}{\frac{2 k}{c}})$

$\Rightarrow t_{1} + t_{2} = \frac{2 h}{c} & t_{1} \cdot t_{2} = \frac{h}{k} . . . . (i)$

Given, $|A B| = 2 d$

$\Rightarrow \sqrt{{(c t_{1} - c t_{2})}^{2} + {(\frac{c}{t_{1}} - \frac{c}{t_{2}})}^{2}} = 2 d$

$\Rightarrow c^{2} {(t_{1} - t_{2})}^{2} + c^{2} {(\frac{1}{t_{1}} - \frac{1}{t_{2}})}^{2} = 4 d^{2}$

$\Rightarrow c^{2} {(t_{1} - t_{2})}^{2} [1 + \frac{1}{{(t_{1} t_{2})}^{2}}] = 4 d^{2}$

$\Rightarrow c^{2} [{(t_{1} + t_{2})}^{2} - 4 t_{1} t_{2}] [1 + \frac{1}{{(t_{1} t_{2})}^{2}}] = 4 d^{2}$

$\Rightarrow c^{2} [\frac{4 h^{2}}{c^{2}} - \frac{4 h}{k}] [1 + \frac{k^{2}}{h^{2}}] = 4 d^{2}$

$\Rightarrow 4 c^{2} h [\frac{h k - c^{2}}{k c^{2}}] [\frac{h^{2} + k^{2}}{h^{2}}] = 4 d^{2}$

$\Rightarrow (h k - c^{2}) (h^{2} + k^{2}) = d^{2} h k$

Replace $h$ by $x$ and $k$ by $y$ .

$\Rightarrow (x y - c^{2}) (x^{2} + y^{2}) = d^{2} x y$

If a rectangular hyperbola have the equation, $x y = c^{2},$ prove that the locus of the middle points of the chords of constant length $2 d$ is $(x^{2} + y^{2}) (x y - c^{2}) = d^{2} x y .$

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

If a rectangular hyperbola have the equation, xy=c2, prove that the locus of the middle points of the chords of constant length 2d is x2+y2xy-c2=d2xy.

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

If a rectangular hyperbola have the equation, $x y = c^{2},$ prove that the locus of the middle points of the chords of constant length $2 d$ is $(x^{2} + y^{2}) (x y - c^{2}) = d^{2} x y .$