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 rectangular hyperbola has double contact with a fixed central conic. If the chord of contact always passes through a fixed point. Prove that the locus of the centre of the hyperbola is a circle passing through the centre of the fixed conic.

Accepted Answer

Let the equation of the fixed conic is $a x^{2} + b y^{2} - 1 = 0$

Now we consider the equation of the conic made by the intersection of the fixed conic and the chord of contact is

$a x^{2} + b y^{2} - 1 + λ {(p x + q y - 1)}^{2} = 0$

Now, this conic equation will be a rectangular hyperbola if,

$a + b + λ (p^{2} + q^{2}) = 0$ $. . . (1)$

And the centre will be found by the equations,

$- a x = λ p (p x + q y - 1)$ $. . . (2)$

and

$- b y = λ q (p x + q y - 1)$ $. . . (3)$

Now if $(h, k)$ , a fixed point on the chord, it satisfies the equation of chord $p x + q y - 1 = 0$

We get ;

$p h + q y = 1$ $. . . . . . (4)$

From equation $(2)$ and $(3)$ , we get ;

$- λ (p x + q y - 1) = \frac{a x}{p}$ and $- λ (p x + q y - 1) = \frac{b y}{q}$

From equation $(4)$ we get ;

$- λ (p x + q y - 1) = a x h + b y k$

Hence $\frac{a x}{p} = a x h + b y k$ and $\frac{b y}{q} = a x h + b y k$

We can derive the values of p, q from the above two equation.

Substitute the values of $p a n d q$ , in equation $(1)$

We get ;

${(a x h + b y k)}^{2} (a + b) + λ (a^{2} x^{2} + b^{2} y^{2}) = 0$

Substitute the values of p and q in equation $(2)$ , we get

${(a x h + b y k)}^{2} + λ (a x^{2} + b y^{2} - a x h - b y k) = 0$

Eliminate $λ$ , we get ;

$(a + b) [a x (x - h) + b y (y - k)] = a^{2} x^{2} + b^{2} y^{2}$

Now this is an equation of a circle, and it passes through the point $(0, 0)$ which is the centre of our conic $a x^{2} + b y^{2} - 1 = 0$

Hence, proved.

A rectangular hyperbola has double contact with a fixed central conic. If the chord of contact always passes through a fixed point. Prove that the locus of the centre of the hyperbola is a circle passing through the centre of the fixed conic.

Important Questions on Hyperbola

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