Mathematics
Equation of a circle
The equation of a circle in the Cartesian coordinate system is given by (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 the coordinate plane, providing a clear relationship between the circle's center, radius, and its equation.
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- 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)
2(x − 3)2 + (y − (−1))2 = (3)2(x − 3)2 + (y + 1)2 = 9The 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 = 49The 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 = 49h = −2 and k = 0center = (−2, 0)The radius is the value of r given the formula (x − h )2 + (y − k )2 = r 2 .Therefore,r 2 = 49r = 7 and −7The 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 = 25To 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 = 4End-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 cm2 . 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. Which of the following is the center of the circle represented by (x − 1)2 + (y + 1)2- eBook - ePub
Dyslexia, Dyscalculia and Mathematics
A practical guide
- Anne Henderson(Author)
- 2013(Publication Date)
- Routledge(Publisher)
Use colour to show the: radius — r, diameter — D, circumference — C.Figure 9.16 Properties of a circleCircle 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.π × D or π × 2r Figure 9.17 How to find the circumference of a circleTo find the area of a circle (answer is in units2 )π × radius × radius which is written πr2 Figure 9.18 How to find the area (A) of a circlePolygons
●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 PolygonsSection 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. The centre is called the point of symmetry, and the shape is described as having rotational symmetry. If you use tracing paper to copy the pattern and turn it around a point it allows a pupil to identify the rotational symmetry of the pattern. - eBook - ePub
- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
.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 O1 A1 ≅ O2 A2 , then O1 ≅ O2 .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.PROBLEMA and B are points on a circle Q such that is equilateral. If the length of side = 12, find the length of arc AB.SOLUTIONTo 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 equilateralGiven the radius, r, and the central angle, n, the arc length is given byTherefore, the length of arc AB = 4π.FORMULAS FOR AREA AND PERIMETERFigures Areas Area (A) of a: square A = s2 ; 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 = πr2 ; where π = 3.14, r = radius sector A = ; where n = central angle, r = radius, π = 3.14 trapezoid A = (h)(b1 + b2 ); where h = height, b1 and b2 = bases Figures Perimeters Perimeter (P) of a: square P = 4s; where s = side rectangle P = 2l + 2w; 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 Points P and R lie on circle Q, m PQR = 120°, and PQ = 18. What is the area of sector PQR - F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
13.5 b. Taking as case study the first possibility – and remembering that F 1 (−c, 0), P eq, 1 (0,b), and C (0,0), one attains (13.75) given the quantitative definition of distance between two given points via their Cartesian coordinates; after revisiting Eq. (13.52) as (13.76) Eq. (13.75) may be redone to (13.77) to eventually reach (13.78) Since b < a due to the definition of a and b as major and minor axes, respectively, 0 < ε < 1 in the case of an ellipse – while ε = 0 is found for a circle, since a = b = R in this case (as plotted above); the eccentricity may accordingly be viewed as a measure of how far the ellipse deviates from being circular. Conversely, a parabola is characterized by ε = 1, and a hyperbola by ε > 1. The ellipse was first studied by Menaechmus (who died 320 BCE), investigated much later by Euclid, but named by Apollonius (who died c. 190 BCE); the concept of foci was introduced by Pappus of Alexandria (290–350 BCE). All conics abide to (13.79) as general Cartesian form – with coefficients being real numbers, and not all A, B, and C being equal to zero (otherwise a straight line would result); which type of conic is at stake depends on the sign of discriminant Δ ≡ B 2 − 4 AC. In fact, Δ < 0 stands for an ellipse, since Eq. (13.53) becomes (13.80) after multiplying both sides by a 2 b 2 and moving a 2 b 2 to the left‐hand side; this leaves A = b 2, B = D = E = 0, C = a 2, and F = −a 2 b 2, as well as Δ = 0 2 − 4 b 2 a 2 = − 4 a 2 b 2 < 0. If A = C = 1, B = D = E = 0, and F = −R 2, then a circle of radius R results, see Eq. (13.79) vis‐à‐vis with Eq. (13.37), again with Δ = 0 2 – 4 = −4 < 0; a parabola is associated with Δ = 0, and a hyperbola with Δ > 0. 13.4 Length of Line Consider a generic curve, y ≡ y { x }, laid on the x 0 y plane, as depicted in Fig. 13.6