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Concept 13 / 14Coordinate Geometry

The Hyperbola: Standard Form, Asymptotes, Conjugate and Rectangular Hyperbolas

The two-foci definition with constant difference 2a, the standard form with b^2 = a^2(e^2 - 1), foci, directrices, latus rectum and focal distances, asymptotes, the conjugate hyperbola, tangents and normals, and the rectangular hyperbola xy = c^2 with its parametric form.

Figure 1
A hyperbola is the locus of a point whose distance from a focus is ee times its distance from a directrix, with e>1e > 1. Equivalently, the difference of its distances from two foci is constant: ∣S′P−SP∣=2a|S'P - SP| = 2a. Standard form: with foci (±ae,0)(\pm ae, 0), x2a2−y2b2=1\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1, where b2=a2(e2−1)b^{2} = a^{2}(e^{2} - 1). Elements: centre (0,0)(0, 0); vertices (±a,0)(\pm a, 0); transverse axis 2a2a (along xx); conjugate axis 2b2b; foci (±ae,0)(\pm ae, 0) with ae=a2+b2ae = \sqrt{a^{2} + b^{2}}; directrices x=±aex = \pm\frac{a}{e} (between the branches); latus rectum 2b2a\frac{2b^{2}}{a}. Unlike the ellipse, bb may be larger or smaller than aa. Focal distances: for PP on the right branch, SP=ex−aSP = ex - a and S′P=ex+aS'P = ex + a, differing by 2a2a. Asymptotes. As ∣x∣→∞|x| \to \infty, y=±bax1−a2x2≈±baxy = \pm\frac{b}{a}x\sqrt{1 - \frac{a^{2}}{x^{2}}} \approx \pm\frac{b}{a}x. So the lines y=±baxy = \pm\frac{b}{a}x, i.e. x2a2−y2b2=0\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 0, are asymptotes: the branches approach them without touching. In the figure they are the diagonals of the 2a×2b2a \times 2b box centred at the origin, the easiest way to sketch a hyperbola. The equation of a hyperbola and of its pair of asymptotes differ only in the constant term, and the angle between the asymptotes is 2tan⁡−1ba2\tan^{-1}\frac{b}{a}. In general, if the asymptotes of a hyperbola are L1=0L_{1} = 0 and L2=0L_{2} = 0, the hyperbola is L1L2=kL_{1}L_{2} = k for some constant kk. Conjugate hyperbola. −x2a2+y2b2=1-\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 has the same asymptotes and the roles of the axes swapped. If e1e_{1} and e2e_{2} are the eccentricities of a hyperbola and its conjugate, then 1e12+1e22=1\frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}} = 1. Parametric form: (asec⁡θ,btan⁡θ)(a\sec\theta, b\tan\theta), using sec⁡2θ−tan⁡2θ=1\sec^{2}\theta - \tan^{2}\theta = 1; or (acosh⁡t,bsinh⁡t)(a\cosh t, b\sinh t) for one branch. Tangents and normals (the ellipse formulas with b2b^{2} replaced by −b2-b^{2}): Tangent at (x1,y1)(x_{1}, y_{1}): xx1a2−yy1b2=1\frac{xx_{1}}{a^{2}} - \frac{yy_{1}}{b^{2}} = 1; at θ\theta: xsec⁡θa−ytan⁡θb=1\frac{x\sec\theta}{a} - \frac{y\tan\theta}{b} = 1. Tangency condition for y=mx+cy = mx + c: c2=a2m2−b2c^{2} = a^{2}m^{2} - b^{2}. Real tangents with slope mm exist only when ∣m∣>ba|m| > \frac{b}{a}, steeper than the asymptotes. Normal at (x1,y1)(x_{1}, y_{1}): a2xx1+b2yy1=a2+b2\frac{a^{2}x}{x_{1}} + \frac{b^{2}y}{y_{1}} = a^{2} + b^{2}. Director circle: x2+y2=a2−b2x^{2} + y^{2} = a^{2} - b^{2}, real only if a>ba > b. Reflection property: the tangent bisects the angle between the focal radii (internally), so rays aimed at one focus reflect towards the other. Rectangular hyperbola. If a=ba = b, the asymptotes y=±xy = \pm x are perpendicular and e=2e = \sqrt{2}. Rotating the axes by 45∘45^{\circ} gives the familiar form xy=c2xy = c^{2}, with the coordinate axes as asymptotes. Its parametric point is (ct,ct)\left(ct, \frac{c}{t}\right). The tangent at tt is x+t2y=2ctx + t^{2}y = 2ct, and the normal is t3x−ty=c(t4−1)t^{3}x - ty = c(t^{4} - 1). The chord joining t1t_{1} and t2t_{2} is x+t1t2y=c(t1+t2)x + t_{1}t_{2}y = c(t_{1} + t_{2}). A tangent to xy=c2xy = c^{2} forms with the axes a triangle of constant area 2c22c^{2}, and the point of contact is the midpoint of the intercepted segment. Key Takeaways: - x2a2−y2b2=1\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1: b2=a2(e2−1)b^{2} = a^{2}(e^{2} - 1), foci (±ae,0)(\pm ae, 0), ∣S′P−SP∣=2a|S'P - SP| = 2a. - Asymptotes y=±baxy = \pm\frac{b}{a}x: the diagonals of the 2a×2b2a \times 2b box. - Hyperbola, asymptotes and conjugate differ only in the constant: 11, 00, −1-1. - 1e12+1e22=1\frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}} = 1 for conjugate hyperbolas. - Tangency: c2=a2m2−b2c^{2} = a^{2}m^{2} - b^{2}; only for ∣m∣>ba|m| > \frac{b}{a}. - Rectangular: xy=c2xy = c^{2}, e=2e = \sqrt{2}, point (ct,ct)(ct, \frac{c}{t}), tangent x+t2y=2ctx + t^{2}y = 2ct. - Pitfall: using the ellipse relation b2=a2(1−e2)b^{2} = a^{2}(1 - e^{2}). Brain-Teaser 1: Find the eccentricity, foci and asymptotes of 16x2−9y2=14416x^{2} - 9y^{2} = 144. Solution: x29−y216=1\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1, so a=3a = 3, b=4b = 4. e=1+169=53e = \sqrt{1 + \frac{16}{9}} = \frac{5}{3}; foci (±5,0)(\pm 5, 0); asymptotes y=±43xy = \pm\frac{4}{3}x, i.e. 16x2−9y2=016x^{2} - 9y^{2} = 0. The conjugate hyperbola y216−x29=1\frac{y^{2}}{16} - \frac{x^{2}}{9} = 1 has e′=54e' = \frac{5}{4}, and 925+1625=1\frac{9}{25} + \frac{16}{25} = 1 as expected. Brain-Teaser 2: Find the hyperbola whose asymptotes are x+2y=3x + 2y = 3 and x−2y=1x - 2y = 1 and which passes through (5,3)(5, 3). Solution: The hyperbola is (x+2y−3)(x−2y−1)=k(x + 2y - 3)(x - 2y - 1) = k. At (5,3)(5, 3): (5+6−3)(5−6−1)=8×(−2)=−16(5 + 6 - 3)(5 - 6 - 1) = 8 \times (-2) = -16. So the hyperbola is (x+2y−3)(x−2y−1)=−16(x + 2y - 3)(x - 2y - 1) = -16. Writing the equation as 'asymptotes product equals constant' uses the key structural fact directly. Brain-Teaser 3: A tangent to xy=c2xy = c^{2} meets the axes at AA and BB. Show that the area of triangle OABOAB is constant and that the point of contact is the midpoint of ABAB. Solution: The tangent at (ct,ct)(ct, \frac{c}{t}) is x+t2y=2ctx + t^{2}y = 2ct. It meets the axes at A(2ct,0)A(2ct, 0) and B(0,2ct)B(0, \frac{2c}{t}). Area =12×2ct×2ct=2c2= \frac{1}{2} \times 2ct \times \frac{2c}{t} = 2c^{2}, independent of tt. The midpoint of ABAB is (ct,ct)(ct, \frac{c}{t}), the point of contact. These two facts are the rectangular-hyperbola analogue of constant subnormal for the parabola.

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The Hyperbola: Standard Form, Asymptotes, Conjugate and Rectangular Hyperbolas — DPP

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