Respuesta :
Equation of circle G: [tex]\bold{(x-3)^{2} + (y-4)^{2} = 49}[/tex]
Equation of circle with center-radius form:
"An equation of a circle with center (p, q) and radius r is:[tex](x-p)^{2}+ (y-q)^{2}=r^{2}[/tex] "
Midpoint Formula:
"The coordinates of the midpoint M of two points (a, b) and (x, y) :
[tex]M(\frac{a+x}{2} ,\frac{b+y}{2} )[/tex] "
Distance formula:
"The distance between two coordinate points (a, b), (m, n):
[tex]d=\sqrt{(n-b)^{2} +(m-a)^{2} }[/tex] "
What is diameter of circle?
"A diameter of a circle is the straight line that from one side of a circle to other side and passes through the center."
For given question,
The line segment connecting the two points B(-4,4) and D (10,4) is a diameter of the circle.
This means, the midpoint of the diameter BD would be the center of a circle G.
Let M represents the center of the circle.
First we find the coordinates of the center of the circle.
Using midpoint formula,
⇒ M = [tex](\frac{-4+10}{2} ,\frac{4+4}{2} )[/tex]
⇒ M = (6/2 , 8/2)
⇒ M = [tex](3,4)[/tex]
Now, we find the distance between points B(-4,4) and D (10,4) to determine the diameter of a circle G.
Using distance formula,
⇒ [tex]d=\sqrt{(4-4)^{2} +(10-(-4))^{2} }[/tex]
⇒ [tex]d=\sqrt{0+(14)^{2} }[/tex]
⇒ [tex]d=14[/tex] units
So, the radius of the circle r would be,
r = d/2
⇒ r = 14 / 2
⇒ r = 7 units
Now, we find the equation of circle with center M(3, 4) and radius r = 7 units.
Using center - radius form equation of circle would be,
[tex](x-3)^{2} + (y-4)^{2} = 7^{2}[/tex]
⇒ [tex]\bold{(x-3)^{2} + (y-4)^{2} = 49}[/tex]
Therefore, option (A) is the correct answer.
Learn more about equation of circle here:
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