Question

Find an equation of the circle that satisfies the stated conditions.(give your answer in standard notation.)
Endpoints of diameter A(4,-9)and B(-6,7)

Answer 1

Answer:

$(x+1)^2+(y+1)^2=89$

Step-by-step explanation:

Equation of a Circle

A circle centered in the point (h,k) and with radius r, can be expressed with the equation:

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

We are given the endpoints of the diameter of a circle as A(4,-9) and B(-6,7).

The center of the circle is the midpoint of segment AB. The midpoint (xm,ym) has coordinates:

$\displaystyle x_m=\frac{4-6}{2}=\frac{-2}{2}=-1$

$\displaystyle y_m=\frac{-9+7}{2}=\frac{-2}{2}=-1$

Center of the circle: (-1,-1)

The radius is half the diameter and the diameter is the distance between the endpoints.

Given two points A(x1,y1) and B(x2,y2), the distance between them is:

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The diameter is:

$d=\sqrt{(-6-4)^2+(7+9)^2}=\sqrt{100+256}=\sqrt{356}=2\sqrt{89}$

The radius is:

$r=2\sqrt{89}/2=\sqrt{89}$

The equation of the circle is:

$(x+1)^2+(y+1)^2=\sqrt{89}^2$

Squaring the root:

$\boxed{(x+1)^2+(y+1)^2=89}$