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# Hyperbola, Equation of Hyperbola, Conic Sections, Co-Ordinate Geometry Assignment Help

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Conic Sections - Hyperbola, Equation of Hyperbola, Conic Sections, Co-Ordinate Geometry

**Hyperbola:** When the ratio (defined in parabola and ellipse) is greater than 1, i.e., e > 1, then the conic is said to be hyperbola.

Since the equation of the hyperbola

differs from that of the a b2, most of the results proved for the ellipse are true for the hyperbola, if we replace b^{2} by - b^{2} in their proofs. We therefore, give below the list of corresponding results applicable in case of hyperbola.

(1) Standard equation of hyperbola is

Where

In this case,

(i) Foci are S (ae, 0) and S' (- ae, 0).

(ii) Equations of directrix ZM ,and Z'M'

(iii) Transverse axis AA' = 2a, conjugate axis BB' = 2b.

(iv) Centre O (0, 0).

(v) Length of latus rectum LL' = L1L1'

(vi) The difference of focal distances from any point P(x1, y1) on hyperbola remains constant and is equal to the length of transverse axis. i.e.,

S'P - SP = (ex1 + a) - (ex1 - a) = 2a.

(2) The equation of rectangular hyperbola x^{2} - y^{2} = a^{2} = b^{2} i.e., in standard form of hyperbola put a = b.

Hence e =

for rectangular hyperbola.

(3) Parametric equations for hyperbola are x = a sec θ and y = b tan θ

(4) Equation of tangent at any point

(x1, y1) on hyperbola is

(5) Equation of tangent in slope form is

y = mx ±

i.e., the line y = mx + c is a tangent to hyperbola,

if c = ±

and point of contact is

(6) the equation of normal at any point (x1, y1) on hyperbola is

(7) Equation of chord of contact at point

(8) Chord whose mid-point is (h, k) is

(9) Chord joining 't1' and 't2' is x + yt1t2 - c(t1 + t2) = 0.

(10) tangent at t1 is x + yt12 - 2ct, = 0

where x1 = ct1,

(11) The length of chord cut off by hyperbola from the line y = mx + c is

(12) Equation of director circle is x^{2} + y^{2} = a^{2} - b^{2}

(13) If Y = m1x and y = m_{2}x are the conjugate diameters of hyperbola, then

(14) Conjugate Hyperbola: The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of the hyperbola is called the conjugate hyperbola of the given hyperbola.

Thus is

= 1 be the given hyperbola, then

conjugate hyperbola.

If e and e' are the eccentricities of these two hyperbolas, then

(15) Equation ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 represents a hyperbola, if Δ ¹ 0, h2 > ab and rectangular hyperbola if Δ ¹ 0, h2 > ab and a + b = 0.

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**Hyperbola:**When the ratio (defined in parabola and ellipse) is greater than 1, i.e., e > 1, then the conic is said to be hyperbola.

Since the equation of the hyperbola

^{2}by - b

^{2}in their proofs. We therefore, give below the list of corresponding results applicable in case of hyperbola.

(1) Standard equation of hyperbola is

Where

(i) Foci are S (ae, 0) and S' (- ae, 0).

(ii) Equations of directrix ZM ,and Z'M'

(iii) Transverse axis AA' = 2a, conjugate axis BB' = 2b.

(iv) Centre O (0, 0).

(v) Length of latus rectum LL' = L1L1'

(vi) The difference of focal distances from any point P(x1, y1) on hyperbola remains constant and is equal to the length of transverse axis. i.e.,

S'P - SP = (ex1 + a) - (ex1 - a) = 2a.

(2) The equation of rectangular hyperbola x

^{2}- y

^{2}= a

^{2}= b

^{2}i.e., in standard form of hyperbola put a = b.

Hence e =

(3) Parametric equations for hyperbola are x = a sec θ and y = b tan θ

(4) Equation of tangent at any point

(x1, y1) on hyperbola is

(5) Equation of tangent in slope form is

y = mx ±

i.e., the line y = mx + c is a tangent to hyperbola,

if c = ±

(8) Chord whose mid-point is (h, k) is

(10) tangent at t1 is x + yt12 - 2ct, = 0

where x1 = ct1,

(11) The length of chord cut off by hyperbola from the line y = mx + c is

(12) Equation of director circle is x

^{2}+ y

^{2}= a

^{2}- b

^{2}

(13) If Y = m1x and y = m

_{2}x are the conjugate diameters of hyperbola, then

(14) Conjugate Hyperbola: The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of the hyperbola is called the conjugate hyperbola of the given hyperbola.

Thus is

If e and e' are the eccentricities of these two hyperbolas, then

^{2}+ 2hxy + by

^{2}+ 2gx + 2fy + c = 0 represents a hyperbola, if Δ ¹ 0, h2 > ab and rectangular hyperbola if Δ ¹ 0, h2 > ab and a + b = 0.

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**Hyperbola**