# Hyperbola - 2

** FOCAL DISTANCES OF A POINT**:

Another Defination of hyperbola is that the difference of focal distances of any point on hyperbola is constant & equal to the length of transverse axis of the hyperbola.

Why ?

Let the hyperbola be

We know that

Distance from focus = (Distance from Directrix)

Hence Distance of point P(x

_{1}, y

_{2}) from S,(ae,0) is

SP = ePM = e = ex

_{1}- a

Similarly S’P = e(PM’) = e = ex

_{1}+ a

Hence S’P - SP = 2a

= Transverse axis

Hence Hyperbola is the Locus of a point which moves in a plane such that the difference of its distances from two fixed points i.e. foci is constant.

__POINT AND HYPERBOLA__The point (x

_{1}, y

_{2}) lies outside, on ,or inside the hyperbola accordingly as < , = or > 0

Why ?

Let P = (x

_{1}, y

_{2}) & Q = (x

_{1}y

_{1})

Draw QL perpendicular to x axis then

QL > PL

y

_{1}> y

_{2}

Adding on both sides

But

Hence

When point lies outside. Similarly we can prove that when point lies on of inside the hyperbola Then

__LINE AND A HYPERBOLA__The line y = mx + c will cut the hyperbola in two points, one point or will not cut accordingly as c

^{2}>, = or < a

^{2}m

^{2}- b

^{2}

Why ?

Let the line be y = mx + c ……………………………………… (1)

& the hyperbola …………………………………… (2)

Eliminating y form (1) & (2) we get

x

^{2}(a

^{2}m

^{2}- b

^{2}) + 2mca

^{2}x + a

^{2}(b

^{2}+ c

^{2}) = 0

This is a quadratic is x, hence

Discriminant = b

^{2}- 4ac

= 4m

^{2}c

^{2}a

^{4}- 4(a

^{2}m

^{2}- b

^{2})(a

^{2})(b

^{2}+ c

^{2})

= c

^{2}+ b

^{2}- a

^{2}m

^{2}

Line will cut in two points if D > 0

c

^{2}+ b

^{2}- a

^{2}m

^{2}> 0 c

^{2}>a

^{2}m

^{2}- b

^{2}

Line will touch the parabola if D = 0

i.e. c

^{2}+ b

^{2}- a

^{2}m

^{2}= 0 c

^{2}= a

^{2}m

^{2}- b

^{2}

Line will not be touch or be touch a chord of parabola

if D < 0 i.e. c

^{2}+ b

^{2}- a

^{2}m

^{2}< 0

c

^{2}< a

^{2}m

^{2}- b

^{2}

**.**

__ILLUSTRATION : 3__For what values of K will the line y = 3x + K be a chord to the hyperbola

Ans: For this hyperbola we have

a

^{2}= 9 & b

^{2}= 45

From the equation of given line

m = 3 & c = k

Hence we know that for a line to be a chord to hyperbola c

^{2}> a

^{2}m

^{2}- b

^{2}

i.e. K

^{2}> 9 x 9 - 45

K

^{2}> 36

K

^{2}- 36 > 0

(K - 6)(K + 6) > 0

Hence

__EQUATIONS OF TANGENT__(a)

__:__

**POINT FORM**The equation of tangent to the hyperbola at (x

_{1}y

_{1}) is

i.e. T = 0 where T =

Why ?

The equation of hyperbola is:

Differentiating w.r.t.x. we get

Equation of tangent when it passes through (x

_{1}y

_{1}) is

(y - y

_{1}) =

a

^{2}y

_{1}y - a

^{2}y

_{1}

^{2}= b

^{2}x

_{1}x - b

^{2}x

_{1}

^{2}

Dividing whole equation by a

^{2}b

^{2}we get

But as (x

_{1}, y

_{1}) lies on hyperbola

is the requurired equation

(b)

**:**

__PARAMETRIC FORM__The equation of tangent to hyperbola at

(a sec, b tan) is

Why ?

We have to paranetric equations of hyperbola as

x = a sec & y = b tan

Differentiating both equations we get dx = a sec tan d & dy = b sec

^{2}d

Hence dividing these equations we get,

Hence the equations of tangent is (y - b tan) =

ay sincos - ab sin

^{2}= cosbx - ab

say sin cos - bx cos = - ab cos

^{2}

is required equation.

**:**

__SLOPE FORM__The equations of tangent to the hyperbola of slope m is y = mx ± & coordinates of points of contact are

Why ?

Let the line with slope m be

y = mx + c, tangent to

Eliminating y from these two equations we get

(a

^{2}m

^{2}- b

^{2})x

^{2}+ 2 mca

^{2}x + a

^{2}(c

^{2}+ b

^{2}) = 0

This is a quadratic equations in x hence for only one solution D should be zero.

D = b

^{2}- 4ac = 4m

^{2}c

^{2}a

^{4}- 4(a

^{2}m

^{2}- b

^{2}(a

^{2}b

^{2}+ c

^{2}) = 0

c

^{2}= a

^{2}m

^{2}- b

^{2}

c =

Hence the equation of tangent is:

y = mx ……………………………. (1)

Now, we also know that the equation of tangent at (x

_{1}, y

_{1}) is

………………………………….. (2)

Comparing (1) & (2) as they are the same equation we get

&

Hence the coordinates of point of contact are .