Respuesta :
Given the points:
(-9, 10) and (-1, 12)
Let's find the equation of the circle using the points which are the endpoints of the diameter.
Apply the general equation for a circle:
[tex](x-h)^2+(x-k)^2=r^2[/tex]Where:
• (h, k) is the center
,• r is the radius of the circle.
Let's first find the diameter of the circle using the distance between points formula:
[tex]d=\sqrt{(x2-x1)^2+(y2-y1)^2}[/tex]Where:
(x1, y1) ==> (-9, 10)
(x2, y2) ==> (-1, 12)
Hence, we have:
[tex]\begin{gathered} d=\sqrt{(-1-(-9))^2+(12-10)^2} \\ \\ d=\sqrt{(-1+9)^2+(12-10)^2} \\ \\ d=\sqrt{(8)^2+(2)^2} \\ \\ d=\sqrt{64+4} \\ \\ d=\sqrt{68} \\ \\ d=8.2 \end{gathered}[/tex]The diameter of the circle is 8.2 units.
To find the radius, we have:
radius = diameter/2 = 8.2/2 = 4.1 units
The radius of the circle is 4.1 units.
Now, let's find the center of the circle.
To find the center of the circle, apply the midpoint formula:
[tex]m=\frac{(x1+x2)}{2},\frac{(y1+y2)}{2}[/tex]Thus, we have:
[tex]\begin{gathered} (h,k)=\frac{-9+(-1)}{2},\frac{10+12}{2} \\ \\ (h,k)=\frac{-9-1}{2},\frac{10+12}{2} \\ \\ (h,k)=\frac{-10}{2},\frac{22}{2} \\ \\ (h,k)=(-5,11) \end{gathered}[/tex]The center of the circle is (-5, 11).
Therefore, the equation of the circle with the points is:
[tex]\begin{gathered} (x-h)^2+(y-k)^2=r^2 \\ \\ (x-(-5))^2+(y-11)^2=4.1^2 \\ \\ (x+5)^2+(y-11)^2=17 \end{gathered}[/tex]ANSWER:
[tex](x+5)^2+(y-11)^2=17[/tex]