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Precalculus: Circles and Distance Formula

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이 집합의 용어 (21)

  • What is the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\)?
    The distance \(d\) is given by \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
  • Define a circle in terms of points and distance.
    A circle is the set of all points in a plane that are at a fixed distance, called the radius, from a fixed point called the center.
  • What is the standard form equation of a circle with center \((h, k)\) and radius \(r\)?
    The equation is \((x - h)^2 + (y - k)^2 = r^2\).
  • Find the center and radius of the circle: \((x - 2)^2 + (y - 3)^2 = 49\).
    Center: \((2, 3)\), Radius: \(7\).
  • Find the center and radius of the circle: \((x + 1)^2 + (y - 4)^2 = 25\).
    Center: \((-1, 4)\), Radius: \(5\).
  • How do you find the standard form of a circle from its general form?
    Complete the square for both \(x\) and \(y\) terms to rewrite the equation in standard form.
  • Convert the general form \(x^2 + y^2 + 6x + 2y + 6 = 0\) to standard form and find center and radius.
    Standard form: \((x + 3)^2 + (y + 1)^2 = 4\), Center: \((-3, -1)\), Radius: \(2\).
  • Convert the general form \(x^2 + y^2 - 10x - 6y - 30 = 0\) to standard form and find center and radius.
    Standard form: \((x - 5)^2 + (y - 3)^2 = 64\), Center: \((5, 3)\), Radius: \(8\).
  • What is the domain and range of the circle \((x - 0)^2 + (y - 0)^2 = 25\)?
    Domain: \([-5, 5]\), Range: \([-5, 5]\).
  • What is the domain and range of the circle \((x + 3)^2 + (y - 2)^2 = 9\)?
    Domain: \([-6, 0]\), Range: \([-1, 5]\).
  • Find the distance between points \((1, -1)\) and \((-4, 11)\).
    Distance is \(13\).
  • Find the distance between points \((6, 7)\) and \((-7, -1)\).
    Distance is \(\sqrt{233}\).
  • Find the distance between points \((2, -6)\) and \((4, -2)\).
    Distance is \(2\sqrt{5}\).
  • Write the standard form of the circle with center \((2, 9)\) and radius 10.
    Equation: \((x - 2)^2 + (y - 9)^2 = 100\).
  • Write the standard form of the circle with center \((-7, 0)\) and radius 5.
    Equation: \((x + 7)^2 + y^2 = 25\).
  • Write the standard form of the circle with center \((-10, 5)\) and radius \(\sqrt{2}\).
    Equation: \((x + 10)^2 + (y - 5)^2 = 2\).
  • Find the center and radius of the circle \((x + 7)^2 + (y - 7)^2 = 16\).
    Center: \((-7, 7)\), Radius: \(4\).
  • Complete the square and write the equation in standard form: \(x^2 - 2x + 1 + y^2 - 4y + 4 = 9\).
    Standard form: \((x - 1)^2 + (y - 2)^2 = 9\), Center: \((1, 2)\), Radius: \(3\).
  • Complete the square and write the equation in standard form: \(x^2 + y^2 + 18x - 10y + 106 = 16\).
    Standard form: \((x + 9)^2 + (y - 5)^2 = 16\), Center: \((-9, 5)\), Radius: \(4\).
  • Complete the square and write the equation in standard form: \(x^2 + y^2 + 14x + 2y = 31\).
    Standard form: \((x + 7)^2 + (y + 1)^2 = 81\), Center: \((-7, -1)\), Radius: \(9\).
  • Complete the square and write the equation in standard form: \(x^2 + y^2 + 12x - 4y + 33 = 0\).
    Standard form: \((x + 6)^2 + (y - 2)^2 = 7\), Center: \((-6, 2)\), Radius: \(\sqrt{7}\).