Question

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)

Answer 1

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

• a) $(x - 5)^2 + (y - 2)^2 = 100$

• b) $(x + 2)^2 + (y + 5)^2 = 169$

• c) $(x - 5)^2 + (y + 1)^2 = 65$

What is the equation of a circle?

• The equation of a circle of radius r and center $(x_0,y_0)$ is given by:

$(x - x_0)^2 + (y - y_0)^2 = r^2$

Item a:

• Center C(5,2), hence $x_0 = 5, y_0 = 2$.

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

$(x - 5)^2 + (y - 2)^2 = r^2$

$(11 - 5)^2 + (10 - 2)^2 = r^2$

$r^2 = 100$

Hence:

$(x - 5)^2 + (y - 2)^2 = 100$

Item b:

• Center C(-2,-5), hence $x_0 = -2, y_0 = -5$.

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

$(x + 2)^2 + (y + 5)^2 = r^2$

$(3 + 2)^2 + (-17 - 5)^2 = r^2$

$r^2 = 169$

Hence:

$(x + 2)^2 + (y + 5)^2 = 169$

Item c:

• Center C(5,-1), hence $x_0 = 5, y_0 = -1$.

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

$(x - 5)^2 + (y + 1)^2 = r^2$

$(-2 - 5)^2 + (-5 + 1)^2 = r^2$

$r^2 = 65$

Hence:

$(x - 5)^2 + (y + 1)^2 = 65$

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