Skip to main content
Indietro

Precalculus: Circles

I pulsanti di controllo sono stati cambiati in modalità "navigazione".
1/20

Termini in questo insieme (20)

  • What is the general equation of a circle?
    The general equation of a circle with center \((h,k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\).
  • How do you find the center and radius from the circle equation \((x - h)^2 + (y - k)^2 = r^2\)?
    The center is \((h,k)\) and the radius is \(r\).
  • What is the standard form of a circle centered at the origin?
    The standard form is \(x^2 + y^2 = r^2\), where \(r\) is the radius.
  • How can you find the radius if given the center and a point on the circle?
    Use the distance formula between the center and the point: \(r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}\).
  • What does the equation \((x - 3)^2 + (y + 2)^2 = 16\) represent?
    A circle with center \((3,-2)\) and radius 4.
  • How do you rewrite the general circle equation \(x^2 + y^2 + Dx + Ey + F = 0\) into standard form?
    Complete the square for both \(x\) and \(y\) terms.
  • What is the geometric meaning of the radius in a circle?
    The radius is the distance from the center to any point on the circle.
  • If a circle has center \((0,0)\) and passes through \((3,4)\), what is its equation?
    The radius is 5, so the equation is \(x^2 + y^2 = 25\).
  • What is the effect of changing \(h\) and \(k\) in the circle equation?
    Changing \(h\) and \(k\) moves the center of the circle horizontally and vertically.
  • How do you determine if a point lies inside, on, or outside a circle?
    Calculate the distance from the point to the center. If less than radius, inside; equal to radius, on; greater than radius, outside.
  • What is the radius of a circle with equation \((x + 1)^2 + (y - 5)^2 = 49\)?
    The radius is 7.
  • How do you find the center of the circle from the equation \(x^2 + y^2 - 6x + 8y + 9 = 0\)?
    Complete the square to get center \((3,-4)\).
  • What is the radius of the circle with equation \(x^2 + y^2 - 6x + 8y + 9 = 0\) after completing the square?
    The radius is 2.
  • What does the term 'locus' mean in relation to a circle?
    A locus is the set of all points equidistant from a fixed point, which defines a circle.
  • How is the distance formula related to the circle equation?
    The circle equation is derived from the distance formula equating the distance from any point to the center to the radius.
  • What is the significance of the constant term in the general circle equation?
    It affects the size and position of the circle after completing the square.
  • How do you graph a circle given its equation in standard form?
    Plot the center, then mark points at a distance equal to the radius in all directions.
  • What is the relationship between the diameter and radius of a circle?
    The diameter is twice the radius: \(d = 2r\).
  • How can you verify if a point lies on a circle given its equation?
    Substitute the point's coordinates into the equation; if true, the point lies on the circle.
  • What is the center and radius of the circle \((x - 2)^2 + (y + 3)^2 = 25\)?
    Center is \((2,-3)\) and radius is 5.