Mathematics ASYMPTOTES OF HYPERBOLA

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ASYMPTOTES OF HYPERBOLA

Asymptotes of Hyperbola :

If the length of the perpendicular let fall from a point on a hyperbola to a straight line tends to zero as the point on the hyperbola moves to infinity along the hyperbola, then the straight line is called the Asymptote of the Hyperbola.

In short asymptote is tangent to hyperbola at infinity.

Let `y = mx + c` is the asymptote of the hyperbola

`x^2/a^2=y^2/b^2=1`.

Solving these two we get the quadratic as

`(b^2- a^2m^2) x^2- 2a^2 mcx - a^2 (b^2 + c^2) = 0` ...... (1)

In order that `y = mx + c` be an asymptote, both roots of equation `(1)` must approach at infinity, the conditions for which are
coeff of `x^2 = 0` & coeff of `x = 0`.

`=> b^2 - a^2m^2 = 0` or `m = pm b/a` & `a^2 mc = 0=> c = 0`.

`:.` equations of asymptote are `x/a+y/b=0` and `x/a-y/b=0`

combined equation to the asymptotes `x^2/a^2-y^2/b^2=0`

As asymptotes of any hyperbola or a curve is a straight line which touches in it two points at infinity.

If the length of the perpendicular let fall from a point on a hyperbola to a straight line tends to zero as the point on the hyperbola moves to infinity along the hyperbola, then the straight line is called the Asymptote of the Hyperbola.

In short asymptote is tangent to hyperbola at infinity.

Let `y = mx + c` is the asymptote of the hyperbola

`x^2/a^2=y^2/b^2=1`.

Solving these two we get the quadratic as

`(b^2- a^2m^2) x^2- 2a^2 mcx - a^2 (b^2 + c^2) = 0` ...... (1)

In order that `y = mx + c` be an asymptote, both roots of equation `(1)` must approach at infinity, the conditions for which are
coeff of `x^2 = 0` & coeff of `x = 0`.

`=> b^2 - a^2m^2 = 0` or `m = pm b/a` & `a^2 mc = 0=> c = 0`.

`:.` equations of asymptote are `x/a+y/b=0` and `x/a-y/b=0`

combined equation to the asymptotes `x^2/a^2-y^2/b^2=0`

As asymptotes of any hyperbola or a curve is a straight line which touches in it two points at infinity.

Properties of Asymptotes :

(i) Perpendicular from the foci on either asymptote meet it in the same points as the corresponding directrix & the common points of intersection lie on the auxiliary circle.

(ii) The angle between the asymptotes of `x^2/a^2-y^2/b^2=1` is `2 tan^(-1)(b/a)`.
and if the angle between the asymptote of a hyperbola `x^2/a^2-y^2/b^2=1` is `2 theta` then `e=sec theta`.

(iii) If from any point on the asymptote a straight line be drawn perpendicular to the transverse axis, the product of the segments of this line, intercepted between the point & the curve is always equal to the square of the semi conjugate axis.

(iv) The tangent at any point `P` on a hyperbola `x^2/a^2-y^2/b^2=1` with centre `C`, meets the asymptotes in `Q` and `R` and cuts off a `Delta CQR` of constant area equal to ab from the asymptotes & the portion of the tangent intercepted between the asymptote is bisected at the point of contact.

(i) Perpendicular from the foci on either asymptote meet it in the same points as the corresponding directrix & the common points of intersection lie on the auxiliary circle.

(ii) The angle between the asymptotes of `x^2/a^2-y^2/b^2=1` is `2 tan^(-1)(b/a)`.
and if the angle between the asymptote of a hyperbola `x^2/a^2-y^2/b^2=1` is `2 theta` then `e=sec theta`.

(iii) If from any point on the asymptote a straight line be drawn perpendicular to the transverse axis, the product of the segments of this line, intercepted between the point & the curve is always equal to the square of the semi conjugate axis.

(iv) The tangent at any point `P` on a hyperbola `x^2/a^2-y^2/b^2=1` with centre `C`, meets the asymptotes in `Q` and `R` and cuts off a `Delta CQR` of constant area equal to ab from the asymptotes & the portion of the tangent intercepted between the asymptote is bisected at the point of contact.

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