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Circles are described below by the coordinates of their centers, C, and one point on their circumference, A. Determine an equation for each circle in center-radius form.

(a) C(5,2) and A(11,10)
(b) C(-2,-5) and A(3,-17)
(c) C(5,-1) and A(-2,-5)

Respuesta :

Using the given center and point at the circumference, the equation of the circles are:

  • a) [tex](x - 5)^2 + (y - 2)^2 = 100[/tex]
  • b) [tex](x + 2)^2 + (y + 5)^2 = 169[/tex]
  • c) [tex](x - 5)^2 + (y + 1)^2 = 65[/tex]

What is the equation of a circle?

  • The equation of a circle of radius r and center [tex](x_0,y_0)[/tex] is given by:

[tex](x - x_0)^2 + (y - y_0)^2 = r^2[/tex]

Item a:

  • Center C(5,2), hence [tex]x_0 = 5, y_0 = 2[/tex].

The point at the circumference is A(11,10), hence:

[tex](x - 5)^2 + (y - 2)^2 = r^2[/tex]

[tex](11 - 5)^2 + (10 - 2)^2 = r^2[/tex]

[tex]r^2 = 100[/tex]

Hence:

[tex](x - 5)^2 + (y - 2)^2 = 100[/tex]

Item b:

  • Center C(-2,-5), hence [tex]x_0 = -2, y_0 = -5[/tex].

The point at the circumference is A(3,-17), hence:

[tex](x + 2)^2 + (y + 5)^2 = r^2[/tex]

[tex](3 + 2)^2 + (-17 - 5)^2 = r^2[/tex]

[tex]r^2 = 169[/tex]

Hence:

[tex](x + 2)^2 + (y + 5)^2 = 169[/tex]

Item c:

  • Center C(5,-1), hence [tex]x_0 = 5, y_0 = -1[/tex].

The point at the circumference is A(-2,-5), hence:

[tex](x - 5)^2 + (y + 1)^2 = r^2[/tex]

[tex](-2 - 5)^2 + (-5 + 1)^2 = r^2[/tex]

[tex]r^2 = 65[/tex]

Hence:

[tex](x - 5)^2 + (y + 1)^2 = 65[/tex]

You can learn more about equation of a circle at https://brainly.com/question/24307696

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