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Basic Concepts of Hyperbola A hyperbola is the locus of a point which moves in a plane in such a way that the ratio of its distance from a fixed point called as focus and a fixed line called the directrix is a constant (called ‘e’ the eccentricity) which is greater than 1. Thus, for hyperbola, the eccentricity e > 1. A hyperbola can be obtained by slicing a right circular cone at both the nappes by a plane. Thus, a hyperbola can be assumed to be a single curve having two branches, one on each nappe. Standard Equation Let S be the focus and ZM the directrix of a hyperbola. Since e > 1, we can divide SZ internally and externally in the ratio e : 1; let the points of division be A and A’ as in the figure. Let AA’ = 2a and be bisected at C. Then, SA = e.AZ, SA’ = e.ZA’ ⇒ SA + SA’ = e(AZ + ZA’) = 2ae i.e., 2SC = 2ae or SC = ae. Similarly by subtraction, SA’ – SA = e(ZA’ – ZA) = 2e.ZC ⇒ 2a = 2eSC ⇒ SC = a/e. Now, take C as the origin, CS as the x-axis, and the perpendicular line CY as the y-axis. Then, S is the point (ae, 0) and ZM the line x = a/e. Let P(x, y) be any point on the hyperbola. Then the condition PS^{2} = e^{2}.(distance of P from ZM)^{2} gives (x – ae)^{2} + y^{2} = e^{2} (x – a/e)^{2} or x^{2}(1 – e)^{2}+ y^{2} = a^{2}(1 – e^{2}) i.e. x^{2}/a^{2} – y^{2}/a^{2}(e^{2}–1) = 1. …… (i) Since e > 1, e^{2} – 1 is positive. Let a^{2} (e^{2} – 1) = b^{2}. Then the equation (i) becomes x^{2}/a^{2} – y^{2}/a^{2} = 1. The eccentricity e of the hyperbola x^{2}/a^{2} – y^{2}/a^{2} = 1 is given by the relation e^{2} = (1 + b^{2}/a^{2}). Since the curve is symmetrical about the y-axis, it is clear that there exists another focus S’ at (–ae, 0) and a corresponding directrix Z’M’ with the equation x = –a/e, such that the same hyperbola is described if a point moves so that its distance from S’ is e times its distance from Z’M’. Terms Related to Hyperbola: Symmetry: Since only even powers of x and y occur in the above equation, hence the curve is symmetrical about both the axis. Foci: S and S’ are the two foci of the hyperbola and their coordinates are (ae, 0) and (-ae, 0) respectively. the distance between the foci is given by SS’ = 2ae. Directrices: ZM and Z’M’ are the two directirces of the hyperbola and their equations are x = a/e and x = -a/e respectively. The distance between the directrices is given by ZZ’ = 2a/e. Axes: The lines AA’ and BB’ are called the transverse axis and conjugate axis respectively of teh hyperbola. the length of the transverse axis is AA’ = 2a. Similarly, teh length of conjugate axis is BB’ = 2b. Centre: The point of intersection of the axes of the hyperbola is called as the centre of the hyperbola. Vertices: The point A(a, 0) and A’(-a, 0) where the curve meets the line joining the foci S and S’ are called the vertices of the hyperbola. Focal Chord: A chord of the hyperbola passing through its focus is called the focal chord. Focal Distance: The focal distance of any point (x, y) on teh hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 are given by (ex-a) and (ex+a). Latus Rectum: If LL’ and NN’ are the latus rectum of the hyperbola tehn these lines are perpendicular to the transverse axis AA’ passing through the foci S and S’ respectively. Then L = (ae, b^{2}/a) , L’ = (ae, -b^{2}/a) , N = (-ae, b^{2}/a) , N’ = (-ae, -b^{2}/a). Hence, the length of latus rectum = LL’ = 2b^{2}/a = NN’. A latus rectum is thus the chord through a focus at right angle to the transverse axis. Important facts:
A hyperbola is the locus of a point which moves in a plane in such a way that the ratio of its distance from a fixed point called as focus and a fixed line called the directrix is a constant (called ‘e’ the eccentricity) which is greater than 1. Thus, for hyperbola, the eccentricity e > 1.
A hyperbola can be obtained by slicing a right circular cone at both the nappes by a plane. Thus, a hyperbola can be assumed to be a single curve having two branches, one on each nappe.
Since e > 1, we can divide SZ internally and externally in the ratio e : 1; let the points of division be A and A’ as in the figure. Let AA’ = 2a and be bisected at C. Then, SA = e.AZ, SA’ = e.ZA’
⇒ SA + SA’ = e(AZ + ZA’) = 2ae
i.e., 2SC = 2ae or SC = ae.
Similarly by subtraction, SA’ – SA
= e(ZA’ – ZA) = 2e.ZC ⇒ 2a = 2eSC ⇒ SC = a/e.
Now, take C as the origin, CS as the x-axis, and the perpendicular line CY as the y-axis. Then, S is the point (ae, 0) and ZM the line x = a/e. Let P(x, y) be any point on the hyperbola. Then the condition PS^{2} = e^{2}.(distance of P from ZM)^{2} gives (x – ae)^{2} + y^{2} = e^{2} (x – a/e)^{2} or x^{2}(1 – e)^{2}+ y^{2} = a^{2}(1 – e^{2})
i.e. x^{2}/a^{2} – y^{2}/a^{2}(e^{2}–1) = 1. …… (i)
Since e > 1, e^{2} – 1 is positive. Let a^{2} (e^{2} – 1) = b^{2}. Then the equation (i) becomes x^{2}/a^{2} – y^{2}/a^{2} = 1.
The eccentricity e of the hyperbola x^{2}/a^{2} – y^{2}/a^{2} = 1 is given by the relation e^{2} = (1 + b^{2}/a^{2}).
Since the curve is symmetrical about the y-axis, it is clear that there exists another focus S’ at (–ae, 0) and a corresponding directrix Z’M’ with the equation x = –a/e, such that the same hyperbola is described if a point moves so that its distance from S’ is e times its distance from Z’M’.
Since only even powers of x and y occur in the above equation, hence the curve is symmetrical about both the axis.
S and S’ are the two foci of the hyperbola and their coordinates are (ae, 0) and (-ae, 0) respectively. the distance between the foci is given by SS’ = 2ae.
ZM and Z’M’ are the two directirces of the hyperbola and their equations are x = a/e and x = -a/e respectively. The distance between the directrices is given by ZZ’ = 2a/e.
The lines AA’ and BB’ are called the transverse axis and conjugate axis respectively of teh hyperbola. the length of the transverse axis is AA’ = 2a. Similarly, teh length of conjugate axis is BB’ = 2b.
The point of intersection of the axes of the hyperbola is called as the centre of the hyperbola.
The point A(a, 0) and A’(-a, 0) where the curve meets the line joining the foci S and S’ are called the vertices of the hyperbola.
A chord of the hyperbola passing through its focus is called the focal chord.
The focal distance of any point (x, y) on teh hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 are given by (ex-a) and (ex+a).
If LL’ and NN’ are the latus rectum of the hyperbola tehn these lines are perpendicular to the transverse axis AA’ passing through the foci S and S’ respectively.
Then L = (ae, b^{2}/a) , L’ = (ae, -b^{2}/a) , N = (-ae, b^{2}/a) , N’ = (-ae, -b^{2}/a).
Hence, the length of latus rectum = LL’ = 2b^{2}/a = NN’.
A latus rectum is thus the chord through a focus at right angle to the transverse axis.
All chords passing through the centre C are bisected at C.
Proof: It can be proved by taking P(x_{1}, y_{1}) as any point on the hyperbola. If (x_{1}, y_{1}) lies on the hyperbola then so does P(–x_{1}, –y_{1}) because the hyperbola is symmetrical about the x and the y axes. Therefore PP’ is a chord whose middle point (0, 0), i.e. the origin C. On account of this property the middle point of the straight line joining the vertices of the hyperbola is called the centre of the hyperbola.
The length of the semi-latus rectum can be obtained by putting x = ae in the equation of the hyperbola. Thus y = b √a^{2}e^{2}/a^{2}–1 = b√e^{2}–1 = b.b/a = b^{2}/a.
The most general equation of hyperbola whose focus is the point (h, k), directrix is lx + my + n = 0 and the eccentricity ‘e’ is given by (x-h)^{2} + (y-k)^{2} = e^{2} (lx + my + n)^{2}/(l^{2} + m^{2})
Since the coordinate x = a sec θ and y = b tan θ satisfy the standard equation x^{2}/a^{2} – y^{2}/b^{2} = 1 for all real values of θ, therefore x = a sec θ and y = b tan θ are the parametric equations of the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1, where the parameter θ lies between 0 ≤ θ ≤ 2π.
A cricle drawn with centre C and transverse axis as a diameter is called the auxiliary circle of the hyperbola. The equation of the auxiliary circle is x^{2} + y^{2} = a^{2}.
The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called the conjugate hyperbola of the given hyperbola. The conjugate hyperbola is therefore of the form
The standard equation of the conjugate hyperbola of the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1 is x^{2}/a^{2} – y^{2}/b^{2} = -1.
Students are advised to refer the coming sections to study these topics in detail.
The line 2x + y = 1 is tangent to the hyperbola x^{2}/a^{2} – y^{2}/b^{2} = 1. If this line passes through the point of intersection of teh nearest directrix and the x-axis then the eccentricity of the hyperbola is ? (IIT JEE 2010)
We substitute (a/e, 0) in the equation y = -2x + 1, we get
0 = -2a/e + 1
This gives a/e = 1/2
Also, y = -2x + 1 is tangent to hyperbola
Hence, 1 = 4a^{2} – b^{2 }
Hence, 1/a^{2 }= 4 – (e^{2} – 1)
Solving this we get, e^{4} – 5e^{2} + 4 = 0
This gives e = 2 or e = 1.
e = 1 gives the conic as parabola. But the conic is given to be a hyperbola so e = 2.
If x = 9 is the chord of contact of the hyperbola x^{2} – y^{2 }= 9, then the equation of the corresponding pair of tangents is ? (IIT JEE 2009)
Let (h, k) be the point whose chord of contact with respect to hyperbola x^{2} – y^{2 }= 9 is x = 9.
We know that the chrod of contact of (h, k) with respect to hyperbola x^{2} – y^{2 }= 9 is T = 0.
Hence, h.x + k(-y) – 9 = 0
This yields h.x - ky – 9 = 0
But it is the equation of the line x = 9.
This is only possible when h = 1 and k = 0.
Again, equation of pair of tangents is T^{2} = SS_{1}.
This gives (x-9)^{2 }=^{ }(x^{2} – y^{2 }- 9) (1^{2} – 0^{2 }- 9)
x^{2} – 18x + 81 = -8x^{2 }+ 8y^{2 }+ 72
This gives 9x^{2} – 8y^{2 }-18x + 9 = 0.
Refer the Useful Books of Mathematics here.
Refer the Revision Notes on Hyperbola for a quick revision.
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