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Circle and parabola in the coordinate system The Coordinate-system: In the Coordinate-system of Descartes the 2 axes are perpendicular to each other. These are called x (abscissa) and y (ordinate) axes. Using this system we can define any point in the plane, only by giving its coordinates (form: P(x;y)). In the same way, we can define the equation of lines or plane figures, like circles or paraboles. The circle Definition of circle: A circle is the set of elements on a plane, which are at equal distance from a point. Equation of a circle when the centre and radius are given Theorem: The equation of a circle in the coordinate-system with centre C(u;v) and radius r: (x-u)2 + (y-v)2 = r2 (Note: if u = 0 and v = 0 (the origin is the centre), then equation is: x2 + y2 = r2) the Proof: Let C(u;v) be centre of the circle, and P(x,y) be the element of the circle. By the definition of circle the length of CP is constant, and equal to r. The distance of points C and P can be written as

follows: Both sides are positive, so raising to the square is an equivalent transformation. (x-u)2 + (y-v)2 = r2 For this reason the solutions of this equation are only those (x;y) pairs of numbers, which are the coordinates of a circle with centre C(u;v) and radius r. Quadratic equations with 2 variables and the circle The equation of the circle can be factorized as follows: (x-u)2+ (y-v)2 = r2 x2-2ux+u2+y2-2vy+v2-r2=0 x2 + y2 – 2ux –2vy + (u2 + v2 – r2) = 0 In general: Ax2+Ay2+Bx+Cy+D=0 where A is the coefficient of x2 and y2, B=-2u, C=-2v and D=(u2 + v2 – r2) (D is constant) For this reason a quadratic equation belongs to a circle if and only if the coefficients of x2 and y2 are equal and the equation has no xy term. Relative position of a circle and a line The number of common points of a circle and a line depends on the number of solutions of the set of equations, consisting of the equation of the circle and that of the line: In case of:  2 solutions: they

intersect each other;  1 solution: the line is the tangent of the circle;  No solution: they do not have a common point. Relative position of 2 circles Similarly to the relative position of a circle and a line:  2 solutions: they intersect each other;  1 solution: they have 1 common point (1 circle can be inside the other);  No solution: no common point (1 can be inside the other). Tangent to a circle from an outer point 1 solution: Using Thales-circle: Given the centre of the circle C(u;v), its radius and an outer point P(x;y), the centre of the Thales circle will be the midpoint of CP, and its radius is (the length of OP)/2. The intersection point of the original circle and the Thales circle are the intersection points of the tangents from P. From these points and the given point P the equation of the 2 tangents can be calculated. The parabola Definition of parabola: A parabola is the set of points on the plane, which are equidistant from a given line – the

directrix – and a given point – the focus. The distance of the focus point and the directrix is called the parameter, the midpoint of the shortest section between the focus and the directrix is the vertex. The equation of the simplest parabola Theorem: If the vertex of a parabola is the origin, its symmetry axis is the y axis and its focus is F(0;p/2), then the equation of this parabola is: x2 = 2py Proof: In this case the equation of the directrix is: y = -p/2 By definition, P(x;y) is the element of the parabola if and only if its distance from the focus and from the directrix is the same: As both sides are positive, raising to the square is an equivalent transformation: x2 +y2 – py + (p/2)2 = y2 + py + (p/2)2 x2 = 2py Q.ed Equation of parabolas with parallel axes to the coordinate axes: • If the parabola x2 = 2py is shifted with vector v(u;v), then its equation will be: (x-u)2 = 2p(y-v) • If the parabola x2 = 2py is rotated about the x axis, then its equation will be:

x2 = -2py • If the parabola x2 = -2py is shifted with vector v(u;v), then its equation will be: (x-u)2 = -2p(y-v) • If the vertex of a parabola is the origin and its focus is F(p/2;0), then its equation is: y2 = 2px • If the parabola y2 = 2px is shifted with vector v(u;v), then its equation will be: (y-v)2 = 2p(x-u) • If the parabola y2 = 2px is rotated about the x axis, then its equation will be y2 = -2px • If the parabola y2 = -2px is shifted with vector v(u;v), then its equation will be (y-v)2 = -2p(x-u) . Quadratic functions and the parabola If a parabola’s axis is perpendicular to the ordinate axis, then its equation can be written as: y = ax2 + bx + c where a=p/2, b=-u/p, c=(u2/2p + v), therefore a, b and c are Real numbers, and a≠0. For this reason, the function of quadratic equations is a parabola, which’s axis is parallel to the y axis. (If the parabola’s axis is parallel to the x axis, then it is not a funtion!) Application: Maths: accurate constructions,

calculations, areas Other: Vectorgraphics, engineering, architecture, physics: v vs. T graph, projectile motion