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Hyperbola Equation Hyperbola Equation In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bows. The hyperbola is one of the four kinds of conic section, formed by the intersection of a plane and a cone. The other conic sections are the parabola, the ellipse, and the circle (the circle is a special case of the ellipse). Which conic section is formed depends on the angle the plane makes with the axis of the cone, compared with the angle a line on the surface of the cone makes with the axis of the cone. If the angle between the plane and the axis is less than the angle between the line on the cone and the axis, or if the plane is parallel to the axis, then the conic is a hyperbola Know More About :- Intro to Precalculus Tutorcircle.com Page No. : 1/4 Hyperbola in daily life can be seen in the form of a decoration or design. It is a smooth curve and an unbounded case of the conic section, formed by the Intersection of a plane with both halves of a double cone. The shape of a Hyperbola is well-defined by its eccentricity e, which is a dimensionless number always greater than one. A hyperbola can also be defined as the locus of all points of the plane whose distances to two fixed points are a constant. Some important terms regarding hyperbola are: Focus of Hyperbola: the two points on the transverse axis that are responsible for the entire shape of the hyperbola. Transverse Axis: the axis on which the two foci lie The standard hyperbola equation is almost identical to the ellipse. If the transverse axis of hyperbola is aligned with the x-axis of a Cartesian coordinate system and is centered at (h, k), the general equation of hyperbola can be written as (x - h)2/A2 - (y - k)2/B2=1 And if the alignment is along the y-axis the equation is given as (y - h)2 / A2 - (x - k)2/B2=1 or (x - h)2/A2 - (y - k)2 / B2= - 1 Where h, k, a and b are real and positive Numbers Example: Show that the line 4p – 3q = 9 touches the hyperbola 4p2 – 9q2 = 27. Solution: For the line y = mx + c to touch the hyperbola x2 / a2 – y2 / b2 = 1, the condition is c2= a2m2 – b2 Here the hyperbola is given as: p2 / (27 / 4) – q2 / (27 / 9) = 1, Learn More :- Dot Products Of Vectors Tutorcircle.com Page No. : 2/4 i.e. here a2 = 27/4, b2 = 27/9 = 3, And comparing the equation of the given line 4p – 3q = 9 with y = mx + c, we get Slope, m = 4/3 and c = –3 ∴ a2m2 – b2 = (27 / 4)(4 / 3)3 – 3 = 12 – 3 = 9 = (–3)2 or a2m2 – b2 = c2, thus satisfying the above condition Hence the given line touches the given hyperbola. Tutorcircle.com Page No. : 3/4 Page No. : 2/3 Thank You For Watching Presentation

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