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

A, B, C

and

D

are the points of intersection of a circle and a rectangular hyperbola. If

A B

passes through the center of the hyperbola, prove that

C D

passes through the center of the circle.

Accepted Answer

Let the equation of the hyperbola be $x y = c^{2}$ and $P$ , be any point on it as $P (c t, \frac{c}{t})$ .
Let the equation to the circle be $x^{2} + y^{2} - 2 h x - 2 k y + λ = 0 . . . . . (1)$ , having center $Q$ $(h, k)$ .
Now if $P$ , lies on $(1)$ , we get $c^{2} t^{2} + \frac{c^{2}}{t^{2}} - 2 h c t - 2 k \frac{c}{t} + λ = 0$ $\Rightarrow t^{4} - \frac{2 h}{c} t^{3} + \frac{λ}{c^{2}} t^{2} - \frac{2 k}{c} t + 1 = 0$ .
If the roots of this equation be $t_{1}, t_{2}, t_{3}$ and $t_{4}$ , then we get $t_{1} + t_{2} + t_{3} + t_{4} = \frac{2 h}{c} . . . . (2)$
$t_{1} t_{2} + t_{1} t_{3} + t_{1} t_{4} + t_{3} t_{4} + t_{2} t_{3} + t_{2} t_{4} = \frac{λ}{c^{2}} . . . . (3)$

$t_{1} t_{2} t_{3} + t_{3} t_{4} t_{1} + t_{4} t_{1} t_{2} + t_{1} t_{2} t_{3} = \frac{2 k}{c} . . . . (4)$

$t_{1} t_{2} t_{3} t_{4} = 1 . . . . (5)$

If $A, B, C$ and $D$ , are the points of intersection of the circle and the hyperbola, there coordinates may be given by $(c t_{1}, \frac{c}{t_{1}}), (c t_{2}, \frac{c}{t_{2}}), (c t_{3}, \frac{c}{t_{3}})$ and $(c t_{4}, \frac{c}{t_{4}})$ .
Equation of line $A B$ , will be $x + y t_{1} t_{2} = c (t_{1} + t_{2}) . . . . (6)$

Since, the line $A B$ passes through the center of the hyperbola $(0,0)$ , we get $t_{1} + t_{2} = 0 . . . . (7)$ .

Similarly, equation of line $C D$ , will be $x + y t_{3} t_{4} = c (t_{3} + t_{4})$ .
If this line passes through center of the circle $(h, k)$ , then $h + k t_{3} t_{4} - c (t_{2} + t_{4}) = 0 . . . . (8)$
From equation $(2)$ and $(7)$ , $t_{3} + t_{4} = \frac{2 h}{c} . . . . . (9)$

And from $(4)$ , $t_{3} t_{4} (t_{1} + t_{2}) + t_{1} t_{2} (t_{4} + t_{3}) = \frac{2 k}{c}$

$t_{1} t_{2} = \frac{k}{h}$

Again, from equation $(5)$ , we get $t_{3} t_{4} = \frac{1}{t_{1} t_{2}} = \frac{h}{k} . . . . (10)$
Putting the values from $(9)$ and $(10)$ in $(8)$ , we get $h + k (\frac{h}{k}) - c (\frac{2 h}{c}) = 0$

Which is identically true.
Hence, $C D$ , passes through the center $(h, k)$ , of the circle.

$A, B, C$ and $D$ are the points of intersection of a circle and a rectangular hyperbola. If $A B$ passes through the center of the hyperbola, prove that $C D$ passes through the center of the circle.

Important Questions on The Hyperbola

Learn and Prepare for any exam you want !

A,B,C and D are the points of intersection of a circle and a rectangular hyperbola. If AB passes through the center of the hyperbola, prove that CD passes through the center of the circle.

Important Questions on The Hyperbola

Learn and Prepare for any exam you want !

$A, B, C$ and $D$ are the points of intersection of a circle and a rectangular hyperbola. If $A B$ passes through the center of the hyperbola, prove that $C D$ passes through the center of the circle.