Question

type the correct answer in each box.
a circle is centered at the point (5, -4) and passes through the point (-3, 2).
the equation of this circle is (x + )² + (y + )² = .

Explanation:

Step1: Recall the standard form of a circle's equation

The standard form of a circle's equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center of the circle. Given the center \((5,-4)\), the equation becomes \((x - 5)^2+(y+4)^2=r^2\).

Step2: Calculate the radius

The radius \(r\) is the distance between the center \((5,-4)\) and the point \((-3,2)\) on the circle. Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Here, \(x_1 = 5,y_1=-4,x_2=-3,y_2 = 2\).
\(r=\sqrt{(-3 - 5)^2+(2+4)^2}=\sqrt{(-8)^2+6^2}=\sqrt{64 + 36}=\sqrt{100}=10\). So \(r^2 = 100\).

Step3: Rewrite the equation in the required form

The equation \((x - 5)^2+(y + 4)^2=100\) can be rewritten as \((x+(- 5))^2+(y + 4)^2=100\).

Answer:

\((x+\boldsymbol{-5})^2+(y+\boldsymbol{4})^2=\boldsymbol{100}\)