Answer :
Answer:
Center: (4,8)
Radius: 2.5
Equation: [tex](x-4)^2+(y-8)^2=6.25[/tex]
Step-by-step explanation:
It was given that; the endpoints of the longest chord on a circle are (4, 5.5) and (4, 10.5).
Note that the longest chord is the diameter;
The midpoint of the ends of the diameter gives us the center;
Use the midpoint formula;
[tex](\frac{x_1+x_2}{2},\frac{y_1+y_2}{2} )[/tex]
The center is at; [tex](\frac{4+4)}{2} ,\frac{5.5+10.5}{2}=(4,8)[/tex]
To find the radius, use the distance formula to find the distance from the center to one of the endpoints.
The distance formula is;
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
[tex]r=\sqrt{(4-4)^2+(10.5-8)^2}[/tex]
[tex]r=\sqrt{0^2+(2.5)^2}[/tex]
[tex]r=\sqrt{0^2+(2.5)^2}=2.5[/tex]
The equation of the circle in standard form is given by;
[tex](x-h)^2+(y-k)^2=r^2[/tex]
We substitute the center and the radius into the formula to get;
[tex](x-4)^2+(y-8)^2=2.5^2[/tex]
[tex](x-4)^2+(y-8)^2=6.25[/tex]
Answer and Step-by-step explanation:
The longest chord on the circle is the diameter, and the center of the circle is the midpoint of the diameter.
Use the midpoint formula to find the center
the x-coordinates are the same, so you can just subtract the y-coordinates to find the radius.
10.5 - 8 = 2.5
The equation of the circle is (x - h)² + (y - k)² = r² where r is the radius and (h, k) is the center.
(x - 4)² + (y - 8)² = 6.25