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Chapter 33. hyperbola. 双曲线. Definition:. The locus of a point P which moves such that the ratio of its distances from a fixed point S and from a fixed straight line ZQ is constant, e , and greater than one .
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Chapter 33 hyperbola 双曲线 hyperbola
Definition: The locusof a point P which moves such that the ratio of its distances from a fixed point Sand from a fixed straight line ZQ is constant, e ,and greater than one. S is the focus, ZQ the directrix and the e, eccentricity of the hyperbola. hyperbola
Simplest form of the eqn of a hyperbola e:eccentricity hyperbola
The foci S, S’ are the points (-ae,0), (ae,0) . Q Q’ y The directrices ZQ, Z’Q’ are the lines x=-a/e, x=a/e . B A’ A x O Z’ S Z S’ AA’ is called the transverse axis=2a . 实轴 B’ BB’ is called the conjugate axis=2b . 虚轴 asymptotes hyperbola
e.g. 1 For the hyperbola , find (i) the eccentricity, (ii) the coordinates of the foci (iii) the equations of the directrices and (iv) the equations of the asymptotes . hyperbola
Soln: (i) hyperbola
(ii) Coordinates of the foci are (iii) Equations of directrices are (iv) Equations of asymptotes are i.e. hyperbola
e.g. 2 Find the asymptotes of the hyperbola . Soln: The asymptotes are hyperbola
e.g. 3 Find the equation of hyperbola with focus (1,1); directrix 2x+2y=1; e= . hyperbola
Soln: From definition of a hyperbola, we have PS=ePM . Where PS is the distance from focus to a point P and PM is the distance from the directrix to a point P. e is the eccentricity of the hyperbola. hyperbola
Let P be (x,y). Hence, distance from P to (1,1) is : Distance from P to 2x+2y-1=0 is : hyperbola
Properties of the hyperbola hyperbola
1. The curve is symmetrical about both axes. The curve exists for all values of y. 2. The curve does not exist if |x|<a. hyperbola
3. At the point (a,0) & (-a,0), the gradients are infinite. 4. Asymptotes of the hyperbola : hyperbola
Many results for the hyperbola are obtained from the corresponding results for the ellipse by merely writing in place of . hyperbola
1. The equation of the tangent to the hyperbola at the point (x’,y’) is 2. The gradient form of the equation of the tangent to the hyperbola is hyperbola
3. The locus of the midpoints of chords of the hyperbola with gradient m is the diameter : hyperbola
e.g. 4 Show that there are two tangents to the hyperbola parallel to the line y=2x-3 and find their distance apart. hyperbola
Soln: Gradient of tangents=2 Hence, equations of tangents are : Perpendicular distance from (0,0) to the lines are : O Distance= hyperbola
The rectangular hyperbola hyperbola
1. A hyperbola with perpendicular asymptotes is a rectangular hyperbola. i.e. b=a So, the standard equation of a rectangular hyperbola is : hyperbola
Equation of a rectangular hyperbola with respect to its asymptotes hyperbola
y y x O x hyperbola
Some simple sketches of the rectangular hyperbola : y y xy=-9 xy=9 x o x o y y x x o o hyperbola
The equation, is satisfied if t is a parameter. The parametric coordinates of any point are : hyperbola
Tangent and normal at the point (ct,c/t) to the curve Gradient of tangent at (ct,c/t) is hyperbola
Equation of tangent at (ct,c/t) is hyperbola
Equation of normal at (ct,c/t) is hyperbola
e.g. 5 The tangent at any point P on the curve xy=4 meets the asymptotes at Q and R. Show that P is the midpoint of QR. hyperbola
Soln: Let P be the point (2t,2/t). Equation of tangent at P is x-axis and y-axis are the asymptotes. When y=0,Q is (4t,0), when x=0 R is (0,4/t). The midpoint of QR is (2t,2/t). hyperbola
Miscellaneous examples on the hyperbola hyperbola
e.g. 6 A chord RS of the rectangular hyperbola subtends a right angle at a point P on the curve. Prove that RS is parallel to the normal at P. hyperbola
Soln: Let S(ct,c/t), R(cp,c/p) and P(cq,c/q). Gradient of tangent at P is : S R P at hyperbola
Gradient of PS= Gradient of RP= Gradient of RS= Hence, hyperbola
Conclusion: In analytic geometry, the hyperbola is represented by the implicit equation : The condition : B2 − 4AC > 0 • (if A + C = 0, the equation represents a rectangular hyperbola. ) Ellipse
In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. The equation will be of the form : Ax2 + Bxy + Cy2 + Dx + Ey + F = 0with A, B, Cnot all zero. hyperbola
then: • if B2 − 4AC < 0, the equation represents an ellipse (unless the conic is degenerate, for example x2 + y2 + 10 = 0); • ifA = C and B = 0, the equation represents a circle; • if B2 − 4AC = 0, the equation represents a parabola; • if B2 − 4AC > 0, the equation represents a hyperbola; • (if A + C = 0, the equation represents a rectangular hyperbola. ) hyperbola
Analyzing an Hyperbola State the coordinates of the vertices, the coordinates of the foci, the lengths of the transverse and conjugate axes and the equations of the asymptotes of the hyperbola defined by 4x2 - 9y2 + 32x + 18y + 91 = 0. hyperbola
~ The end ~ hyperbola
Ex 14d do Q1, 3, 5, 7, 9, 11. Misc.14 no need to do hyperbola
Ex 14d Q 1 At point (2a,a/2), Gradient of normal at (2a,a/2) is 4. hyperbola
Equation of normal at (2a,a/2) is : hyperbola
Ex 14d Q 3 (2t,2/t) 2y=x+7 4/t=2t+7 hyperbola
At point A, t=1/2 so, A is (1,4) At point B, t=-4 so, B is (-8,-1/2) x=2t, y=2/t Hence, xy=4 hyperbola
At point A, dy/dx=-4 At point B,dy/dx=-1/16 Eqn of tangent at A : hyperbola