#Geometry the same line m1 and m2 Pc#

Equation of a Circle An equation of the circle with center and radius r is (y r 2 In particular, if the center is the origin the equation is x2 y2 r2 For instance, an equation of the circle with radius 3 and center is (y 9 Stewart: Calculus, Sixth Edition. From the distance formula, we have y P (x, y) r C (h, k) y r or equivalently, squaring both sides, we get 0 FIGURE 5 (y r 2 x This is the desired equation.

#Geometry the same line m1 and m2 Pc#

(See Figure 5.) Thus, P is on the circle if and only if PC r. use the distance formula to find the equation of a circle with radius r and center definition, the circle is the set of all points whose distance from the center is r.

(See Figure 4.) To find the distance P1 P2 between any two points 1, y1 and P2 2, y2 we note that triangle P1P2 P3 in Figure 4 is a right triangle, and so the Pythagorean Theorem we have x P P P P FIGURE 4 1 2 1 3 2 P2 P3 2 s x2 x1 2 y2 y1 2 2 x 1 y1 Distance Formula The distance between the points 1, y1 and P2 2, y2 is P P 1 2 2 x 1 y1 For instance, the distance between and is 2 2 s4 2 5 2 s41 An equation of a curve is an equation satisfied the coordinates of the points on the curve and no other points. Thus, the distance between points 1, y1 and P3 2, y1 on a horizontal line must be x 2 x 1 and the distance between P2 2, y2 and P3 2, y1 on a vertical line must be y2 y1. Thus, the region consists of all points that lie between (but not on) the horizontal lines y 1 and y lines are shown as dashed lines in Figure 3(c) to indicate that the points on these lines lie in the y 0 Recall from Review of Algebra that the distance between points a and b on a number line is a b b a. y y y 0 x 0 x 0 x FIGURE 3 (a) x 0 (b) (c) y 1 2 REVIEW OF ANALYTIC GEOMETRY (b) The set of all points with 1 is a horizontal line one unit above the Figure (c) Recall from Review of Algebra that 1 y 1 if and only if The given region consists of those points in the plane whose lie between and 1. (a) x (c ) (b) y y SOLUTION (a) The points whose are 0 or positive lie on the or to the right of it as indicated the shaded region in Figure 3(a). EXAMPLE 1 Describe and sketch the regions given the following sets. Notice that the first quadrant consists of those points whose and are both positive. The and are called the coordinate axes and divide the Cartesian plane into four quadrants, which are labeled I, II, and IV in Figure 1. The plane supplied with this coordinate system is called the coordinate plane or the Cartesian plane and is denoted 2. Often we identify the point P with the ordered pair and refer to point the notation used for an open interval is the same as the notation used for a point you will be able to tell from the context which meaning is This coordinate system is called the rectangular coordinate system or the Cartesian coordinate system in honor of the French mathematician Descartes even though another Frenchman, Pierre Fermat invented the principles of analytic geometry at about the same time as Descartes. 2 1 1 _3 _2 _1 O _1 _2 (1, 3) 3 2 3 4 5 x (2, _4) FIGURE 2 reversing the preceding process we can start with an ordered pair and arrive at the corresponding point P.

We say that P is the point with coordinates and we denote the point the symbol Several points are labeled with their coordinates in Figure 2. Then the point P is assigned the ordered pair The first number a is called the of P the second number b is called the of P. Draw lines through P perpendicular to the and These lines intersect the axes in points with coordinates a and b as shown in Figure 1. Usually one line is horizontal with positive direction to the right and is called the the other line is vertical with positive direction upward and is called the Any point P in the plane can be located a unique ordered pair of numbers as follows. We start drawing two perpendicular coordinate lines that intersect at the origin O on each line. For example, let's take the x-axis ( y = 0) as the first line, and the line defined by y = x – 1 as the second line.įor the second line, the slope m is equal to 1 which means 45 degrees, and the y-intercept is equal to -1 which means that the line intercepts the y-axis in the point (0, -1).Voorbeeld tekst REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers.