Equation of Circles
The equation of a circle in the Cartesian coordinate system is given by the formula (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the coordinates of the center of the circle and r is the radius. This equation allows for the representation and analysis of circles in a two-dimensional plane, providing a fundamental tool in geometry and algebra.
5 Key excerpts on "Equation of Circles"
eBook - ePub- Richard C. Dorf, Ronald J. Tallarida(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...A circle at center O with radius c contains the vertices and illustrates the relation among a, b, and c. Asymptotes have slopes b / a and − b / a for the orientation shown. FIGURE 4.12. Hyperbola with center at (h, k): (x − h) 2 a 2 − (y − k) 2 b 2 = 1 ; slopes of asymptotes ± b / a. FIGURE 4.13. Hyperbola with center at (h, k): (y − k) 2 a 2 − (x − h) 2 b 2 = 1 ; slopes of asymptotes ± a / b. If the focal axis is parallel to the x-axis and center (h, k),. then (x − h) 2 a 2 − (y − k) 2 b 2 = 1 9. Change of Axes A change in the position of the coordinate axes will generally change the coordinates of the points in the plane. The equation of a particular curve will also generally change. • Translation When the new axes remain parallel to the original, the transformation is called a translation (Figure 4.14). The new axes, denoted x ′ and y ′ have origin 0′ at (h, k) with reference to the x and y axes. FIGURE 4.14. Translation of axes. FIGURE 4.15. Rotation of axes. A point P with coordinates (x, y) with respect to the original has coordinates (x ′, y ′) with respect to the new axes. These are related by x = x ′ + h y = y ′ + k For example, the ellipse of Figure 4.10 has the following simpler equation with respect to axes x ′ and y ′ with the center at (h, k): y ′ 2 a 2 + x ′ 2 b 2 = 1. • Rotation When the new axes are drawn through the same origin, remaining mutually perpendicular, but tilted with respect to the original, the transformation is one of rotation. For angle of rotation ϕ (Figure 4.15), the coordinates (x, y) and (x ′, y ′) of a point P are related by x = x ′ cos ϕ − y ′ sin ϕ y = x ′ sin ϕ + y ′ cos ϕ 10. General Equation of Degree Two A x 2 + B x y + C y 2 + D x + E y + F = 0 Every equation of the above form defines a conic section or one of the limiting forms of a conic. By rotating the axes through a particular angle ϕ, the xy -term vanishes, yielding A ′ x ′ 2 + C ′ y ′ 2 + D ′ x ′ + E ′ y ′ + F ′ = 0 with respect to the axes x ′ and y ′...
- Rebecca Dayton(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...(y − k) 2 = r 2 (x − 3) 2 + (y − (−1)) 2 = (3) 2 (x − 3) 2 + (y + 1) 2 = 9 The equation of the circle is (x − 3) 2 + (y + 1) 2 = 9. Example: Determine the center and the radius of the circle. (x + 2) 2 + y 2 = 49 The center is located at the point (h, k) given the formula (x − h) 2 + (y − k) 2 = r 2. Therefore, (x + 2) 2 + y 2 =. 49 (x − (−2)) 2 + (y − 0) 2 = 49 h = −2 and k = 0 center = (−2, 0) The radius is the value of r given the formula (x − h) 2 + (y − k) 2 = r 2. Therefore, r 2 = 49 r = 7 and −7 The radius is 7 because length has to be a positive value. The center is (−2, 0) and the radius is 7. Example: Graph x 2 + (y − 3) 2 = 25 To graph the circle, determine the center and the radius. Center: (h, k) = (0, 3) Exercise 4 1. Write the equation of a circle with center (−1, −5) and a radius that measures 12 units. 2. Determine the center and the radius of the circle. 3. Graph (x + 2) 2 + (y − 1) 2 = 4 End-of-Chapter Quiz Identify each of the following as a radius, chord, diameter, secant, or tangent. 1. 2. 3. 4. 5. 6. 7. Find. A. 80° B. 140° C. 160° D. 180° 8. Find m∠LPM. A. 35° B. 55° C. 70° D. 110° 9. Find. A. 20° B. 25° C. 30° D. 35° 10. The area of a circle is approximately 153.86 cm 2. Find the circumference of the circle. 11. Chloe the Clown rides a unicycle. The diameter of the unicycle wheel is 20 inches. Approximately how many times will the wheel turn to ride 10 feet? 12. Margaret wants to plant grass in her backyard. The bags of seed at the store state that one bag covers 150 square feet. If the semicircle below represents Margaret’s backyard, how many bags of seed will she need to buy? 13. Find the length of. 14. Find the area of sector BOC. 15. The figure shows 12 equally space points around the circle. What is the length of ? 16. Anika’s paper fan is a sector with radius 6 inches and a central angle of 160°. What is the area of Anika’s fan? 17...
eBook - ePubDyslexia, Dyscalculia and Mathematics
A practical guide
- Anne Henderson(Author)
- 2013(Publication Date)
- Routledge(Publisher)
...the: radius — r, diameter — D, circumference — C. Figure 9.16 Properties of a circle Circle facts ● The circumference is the perimeter of a circle. ● The diameter is twice the length of the radius. ● The radius is half the length of the diameter (divide D by 2). ● Pi π (pronounced pie) is important. ● Pi π has a value of 3.142. ● Press EXP on the calculator to use π. ● A 3D shape with circular top and bottom is a cylinder. To find the circumference of a circle: (answer is in units) π × D or π × 2r A rhyme to help: Fiddle de-dum, Fiddle de-dee The ring round the moon is π times D. Figure 9.17 How to find the circumference of a circle To find the area of a circle (answer is in units 2) π × radius × radius which is written πr 2 A rhyme to help: A round hole in my sock Has just been repaired. The area mended Is pi r squared. Figure 9.18 How to find the area (A) of a circle Polygons ● Copy, cut out, stick onto card and turn the angle pictures and facts given into a memory card. (number 23, see page 144). ● Multi-sided figures are generally called polygons. They have individual names depending on the number of sides, but many students find these difficult to remember. Figure 9.19 Polygons Section E: Co-ordinates The two straight lines at right angles to each other on a graph are called the axes. Coordinates are a pair of numbers, usually in brackets, which describe the precise location of a point on the axes. The one which is horizontal is called the x -axis (because x is a cross) and the vertical line is called the y -axis. The first number indicates the x -axis value (across the hall) and the second number indicates the y -axis value (up the stairs). For example: (3, 5) means 3 units across to the right and 5 units up. Figure 9.20 Graph to show the position of co-ordinate (3,5) Section F: Rotational symmetry This is the description given when a pattern is rotated around a point to identify the number of times the pattern is repeated...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...The distance of point P to the diagrammed circle with center O is the line segment, part of line segment. A line that has one and only one point of intersection with a circle is called a tangent to that circle, and their common point is called a point of tangency. In the diagram, Q and P are each points of tangency. A tangent is always perpendicular to the radius drawn to the point of tangency. Congruent circles are circles whose radii are congruent. If O 1 A 1 ≅ O 2 A 2, then O 1 ≅ O 2. Circles that have the same center and unequal radii are called concentric circles. A circumscribed circle is a circle passing through all the vertices of a polygon. The polygon is said to be inscribed in the circle. PROBLEM A and B are points on a circle Q such that is equilateral. If the length of side = 12, find the length of arc AB. SOLUTION To find the length of arc AB, we must find the measure of the central angle AQB and the measure of radius. AQB is an interior angle of the equilateral triangle. Therefore, m AQB = 60°. Similarly, in the equilateral Given the radius, r, and the central angle, n, the arc length is given by Therefore, the length of arc AB = 4π. FORMULAS FOR AREA AND PERIMETER Figures Areas Area (A) of a: square A = s 2 ; where s = side rectangle A = lw ; where l = length, w = width parallelogram A = bh ; where b = base, h = height triangle A bh ; where b = base, h = height circle A = πr 2 ; where π = 3.14, r = radius sector A = ; where n =. central angle, r = radius, π = 3.14 trapezoid A = (h)(b 1 + b 2); where h = height, b 1 and b 2 = bases Figures Perimeters Perimeter (P) of a: square P = 4 s ; where s = side rectangle P = 2 l + 2 w ; where l = length, w = width triangle P = a + b + c ; where a, b, and c are the sides Circumference (C) of a circle C = πd ; where π = 3.14, d =. diameter PROBLEM Points P and R lie on circle Q, m PQR = 120°, and PQ = 18...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...(13.51) to (13.53) usually known as canonic equation of an ellipse. If Eq. (13.37) is rewritten as (13.54) following division of both sides by R 2, then one confirms – upon comparison with Eq. (13.53), that a circumference is a particular case of an ellipse, corresponding specifically to a = b = R. Obviously that Eq. (13.53) may be solved for y 2 as (13.55) or, after taking square roots of both sides, (13.56) the plus sign corresponds to the upper portion (P md, 1 P eq, 1 P md, 2) of the ellipse, and the minus sign to its lower counterpart (P md, 1 P eq, 2 P md, 2) in Fig. 13.5 b. In the particular case of point P eq being equidistant from F 1 and F 2 as outlined in Fig. 13.5 b, one may resort to Eq. (13.38) to write (13.57) whereas point (x, y) being equidistant F 1 and F 2 enforces (13.58) one may then take squares of both sides to get (13.59) while dropping y 2 from both sides yields (13.60) Square roots are now to be taken of both sides, thus leaving Eq. (13.60) as (13.61) which is equivalent to (13.62) the first condition in Eq. (13.62) can never be fulfilled unless c = 0 (in which case a circumference with coincident foci at its center, rather than an ellipse would be at stake). Conversely, the second condition leads to (13.63) following cancellation of terms alike between sides, or merely (13.64) Therefore, the points on the plane equidistant from the two foci lie on a vertical straight line passing through the origin, so combination of Eqs. (13.56) and (13.64) produces (13.65) this means that constant b, defined by Eq. (13.52), is but the distance between C (0,0) and either P eq, 1 or P eq, 2 on the ellipse, i.e. (13.66) as highlighted in Fig. 13.5 b – so the distance between P eq, 1 and P eq, 2 should read (13.67) or, in view of Eq. (13.66), (13.68) Eq...
