Question

Find the center and radius of the circle. Write the standard form of theequation.(2,5)(5,5)

Answer 1

If (2.5) and (5,5) are the two end points of the circle, then the center of the circle is the midpoint

To find the midpoint, we will use the formula

$(x_m,y_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

x₁ = 2 y₁=5

x₂=5 y₂=5

Substitute into the formula and evaluate

$(x_m,y_m)=(\frac{2+5}{2},\frac{5+5}{2})$$=(\frac{7}{2},\frac{10}{2})$$=(\frac{7}{2},5)$

Hence, the center of the circle is (7/2 , 5)

To find the radius, we can simply find the distance between the end points and divide by 2

Using the distance formula,

$|d|=\sqrt[]{(5-2)^2+(5-5)^2}$$=\sqrt[]{3^2}=3$

Hence, the radius is 3

The general formula for the equation of a circle centered (h,k) and radius r is given by

$(x-h)^2+(y-k)^2=r^2$

substitute the values into the formula

$(x-\frac{7}{2})^2+(y-5)^2=3^2$

Therefore, the equation of the circle is:

$(x-\frac{7}{2})^2+(y-5)^2=9$