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Students are advised to remember all the important properties of hyperbola on their fingertips so as to ace the competitions like the JEE with ease. Section 9.2 The Hyperbola 887 Use the distance formula. Graphing and Properties of Hyperbolas Date_____ Period____ Identify the vertices, foci, and direction of opening of each. It is the locus of a point that moves in such a way that the ratio of its distance from a fixed point (called Focus) to its distance from a fixed line (called Directrix) equals to constant ‘e’ which is greater A level cut gives a circle, and a moderate angle produces an ellipse. The curves are "conic sections."
If e > 1, the graph is an hyperbola.
If e = 1, the graph is a parabola. ELLIPSE, HYPERBOLA AND PARABOLA ELLIPSE Concept Equation Example Ellipse with Center (0, 0) Standard equation with a > b > 0 Horizontal major axis: Vertical major axis Center (0, 0); major axis: vertical Vertices: (0, ±3); foci: ( 0, ± √5) (c2 = a 2 – b2 = 9 – 4 = 5, so c = √5.) A hyperbola has two pieces, called connected A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bows. 1) x2 81 − y2 4 = 1 2) x2 121 − y2 81 = 1 3) y2 25 − x2 16 = 1 4) x2 121 − y2 36 = 1 5) (x + 2)2 169 − (y + 8)2 4 = 1 6) (y + 8)2 36 − (x + 2)2 25 = 1-1-©3 v2g0 i1I1 N vK kuRtja q RSto LfvtaweaPrwew BLSLQCZ.R i WAElLlt ZrMi1gVhMtTs X mrSeDs6e 3rQvmeld9.
A hyperbola is the set of points in a plane the difference of whose distances from two fixed points, called foci, is constant. hyperbola the difference of the distances between the foci and a point on the hyperbola is fixed. Important Properties of Hyperbola . A steep cut gives the two pieces of a hyperbola (Figure 3.15d). The parabola and ellipse and hyperbola have absolutely remarkable properties.
The Greeks discovered that all these curves come from slicing a cone by a plane. Vertex Vertex x y Transverse axis Focus Center Focus Figure 9.14 The two branches of a hyperbola Figure 9.15 St. Mary’s Cathedral P-BLTZMC09_873-950-hr 21-11-2008 13:28 Page 886.
The line segment connecting the vertices is the transverse axis, and the midpoint of the transverse axis is the center of the hyperbola. (If e = 0, the graph is a circle.) Hyperbola is one of the conic sections. e > 1. 9.6 Properties of the Conic Sections Contemporary Calculus 5 For e ≥ 0, the polar coordinate graphs of r = k 1 ± e.cos(θ) and r = k 1 ± e.sin(θ) are conic sections with one focus at the origin. Now, we know that hyperbola is a conic whose eccentricity is greater than unity i.e.
FIGURE 10.29 FIGURE 10.30 The graph of a hyperbola has two disconnected branches. hyperbola is specifically a smooth curve that lies in a plane.
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