Name: Conjugate Hyperbola
Uploaded: 2019-07-19
Duration: 12 min 19 s
Description: In this video, we will learn about the transverse axis and conjugate axis of any hyperbola with respect to another hyperbola. Watch and learn.

Question

Show that the equation to the director circle of the conic

x y = c^{2}

is

x^{2} + 2 x y \cos ω + y^{2} = 4 c^{2} \cos ω

.

Accepted Answer

The equation of the hyperbola is given as,

$x y = c^{2} \dots (1)$

The equation of any tangent at any point $t$ is given by,

$x y_{1} + x_{1} y = 2 c^{2}$

$\Rightarrow x \frac{c}{t} + c t y = 2 c^{2}$

$\Rightarrow t^{2} y + x = 2 c t$

If $t$ is any point on it, then the equation of the tangent at $t$ is given as,

$t_{1}^{2} y + x = 2 c t_{1} \dots (2)$

Similarly, the equations of the tangent at any other point $t_{2}$ will be given as,

$t_{2}^{2} y + x = 2 c t_{2} \dots (3)$

Solving equations $(2)$ and $(3)$ , we get,

$x = \frac{2 c t_{1} t_{2}}{t_{1} + t_{2}}$

$y = \frac{2 c}{t_{1} + t_{2}}$

Hence, we get,

$t_{1} t_{2} = \frac{x}{y} \dots (4)$

$(t_{1} + t_{2}) = \frac{2 c}{y} \dots (5)$

Now, if the tangents given by $(2)$ and $(3)$ are perpendicular to one another, and $ω$ is the angle at which the axes are inclined, then we get,

$1 + (\frac{- 1}{t_{1}^{2}} - \frac{1}{t_{2}^{2}}) \cos ω + \frac{1}{t_{1}^{2}} . \frac{1}{t_{2}^{2}} = 0$

$1 - \frac{(t_{1}^{2} + t_{2}^{2}) \cos ω}{t_{1}^{2} t_{2}^{2}} + \frac{1}{t_{1}^{2} t_{2}^{2}} = 0$

$t_{1}^{2} t_{2}^{2} - [{(t_{1} + t_{2})}^{2} - 2 t_{1} t_{2}] \cos ω + 1 = 0 \dots (6)$

Substituting the values of $t_{1} t_{2}$ and $(t_{1} + t_{2})$ from equations $(4)$ and $(5)$ , we get the locus of the point of intersection of the two perpendicular tangents as,

$\frac{x^{2}}{y^{2}} - (\frac{4 c^{2}}{y^{2}} - \frac{2 x}{y}) \cos ω + 1 = 0$

$x^{2} - (4 c^{2} - 2 x y) \cos ω + y^{2} = 0$

$x^{2} + y^{2} + 2 x y \cos ω = 4 c^{2} \cos ω$

Hence proved.

Show that the equation to the director circle of the conic $x y = c^{2}$ is $x^{2} + 2 x y \cos ω + y^{2} = 4 c^{2} \cos ω$ .

Important Questions on The Hyperbola

Learn and Prepare for any exam you want !

Show that the equation to the director circle of the conic xy=c2 is x2+2xycosω+y2=4c2cosω.

Important Questions on The Hyperbola

Learn and Prepare for any exam you want !

Show that the equation to the director circle of the conic $x y = c^{2}$ is $x^{2} + 2 x y \cos ω + y^{2} = 4 c^{2} \cos ω$ .