BodhiAI

Nov. 13, 2019

Hyperbola

A hyperbola is the locus of a point which moves in a plane such that its distance from a fixed point i.e. focus always bears a constant ratio to its distance from a fixed line i.e. directrix.

Mathematically,

where, e is called eccentricity or contracity and is equal to the ratio of distances of any point from the focus and the directrix. For a hyperbola e is greater than unity i.e. e > 1.

Important

A hyperbola has two foci and pair of directrices like ellipse.

Equation of Hyperbola in Standard Form

The equation of hyperbola in standard form is given by the equation

such that, the intercept along x-axis is (put y = 0)

**⇒ x ^{2} = a^{2}**

^{∴ x = ±a}

i.e. the hyperbola intersects the x-axis in points (a, 0) and (-a, 0)

Also, the intercepts along y-axis is (put x = 0)

which is imaginary, hence the curve (hyperbola) does not intersect y-axis. Also replacing x by -x and y by -y in the equation of the hyperbola, the equation remains the same. Hence, the hyperbola is symmetric in x as well as y-axis. Thus, the hyperbola can be graphically represented as

Coordinates of Focii and Equation of Directrices

Since, the hyperbola has two foci and a pair of corresponding directrices. So, according to mathematical interpretation of hyperbola, we get

Now, if P approaches vertex A, then M will approach Z and M' approaches Z', thus

i.e. the point A divides the join of SZ internally in the ratio e : 1 and that of S'Z' externally in the ratio e : 1.

Important

The vertex divides the join of focus and the point of intersection of directrix with axis internally and externally in the ratio of e:1.

Now, SA + SA' = e(AZ + A'Z)

(CS-CA)+ (CS+CA) = e AA'

2CS = e(2a) {AA' = 2a}

CS = ae = CS'

Hence, the coordinates of foci are

S (ae, 0) and S' (-ae, 0)

Also, SA' - SA = e(A'Z-AZ)

AA'=e(CA' + CZ) - (CA-CZ))

2a = e(CA' + CZ - CA + CZ) {CA' = CA}

2 eCZ = 2a

CZ =

Hence, the equation of directrices are

Important

Inferences for the standard equation of Hyperbola.

Let the standard equation of the Hyperbola is of the form

**Symmetry Replacing x by -x and y by -y, the equation of the curve remains unchanged. Hence, the curve is symmetric in both x as well as y-axis i.e. the curve is symmetric in the xy plane.**

**Since, domain of the hyperbolic function lies outside -a and a. So no part of the curve can be made between -a and a. Also, the range y R i.e. the curve can output any value in between -∞ and +∞.**

Origin Origin is the centre of the Hyperbola.

Intersection The curve meets the co-ordinate axes at x = -a and x = +a and no part of the curve

With Axes intersect the y-axis.

Some Terms Related to Hyperbola

Let is the standard equation of the hyperbola, having focil S and S' at and are the eqution of the directrices. Then for such a hyperbola, the following important terms can be defined.

Axis of the conic section is the line through the focus and perpendicular to the directrix.

For a hyperbola, there are two axis, one being the Transverse Axis and the other being the Conjugate Axis.

Axis TRANSVERSE AXIS is the one which lie along the line passing through the foci and perpendicular to the directrices and CONJUGATE AXIS is the one which is perpendicular to the transverse axis and passes through the midpoint of the foci.

Important

In the equation, since the -iv sign is with y part, hence y-axis is CONJUGATE and thus x-axis is TRANSVERSE axis.

On the other hand, if the equation is of the form , here the -iv sign is with x part, hence x is CONJUGATE and y is TRANSVERSE AXIS. Such a hyperbola is called CONJUGATE HYPERBOLA.

Note: In both the cases, the value on the RHS i.e. 1 must be positive.

Vertex The points of intersection of the curve with the axis are the vertices of the conic.

For a hyperbola, the x-axis is TRANSVERSE axis, hence the vertices are A(a, 0) and A'(-a, 0).

For a Conjugate hyperbola, the y-axis is TRANSVERSE axis, hence the vertices are B (0, b) and B' (0, -b).

Centre Since, all chords passing through C is bisected at C (as C is the centre of symmetry). Hence, C(0, 0) is the centre of hyperbola.

Through the point on the hyperbola, a perpendicular drawn to the axis of the hyperbola such

Double that it meets the other end of the symmetric curve at . Then the

Ordinate line OQ' is called the Double Ordinate.

Thus, Length of Double Ordinate = Double (Ordinate part)

Latus Rectum The double ordinate passing through focus is called Latus Rectum. Thus, Length of Latus Rectum

Problem Solving Trick Length of Latus Rectum

Focal Chord The chord of the hyperbola through the focus is called a focal chord.

Distance of the point P on the hyperbola from the focus is called focal distance.

Focal Distance

Focal Distance |SP| =(ex_{1}- a)

Also,