Question

the length from the center to any point on the circle is the

to find the center of a circle from the equation ((x - h)^2+(y - k)^2=r^2), identify the values of

the circle described by ((x - 3)^2+(y + 5)^2 = 25) has a radius of

in the equation ((x + 1)^2+(y - 4)^2 = 36), the circles center is located at

the formula ((x - h)^2+(y - k)^2=r^2) is used to represent the
of a circle.

Explanation:

Step1: Recall circle - radius definition

The length from the center to any point on the circle is the radius.

Step2: Recall circle - center formula

For the equation of a circle \((x - h)^2+(y - k)^2=r^2\), the center is \((h,k)\), so we identify \(h\) and \(k\).

Step3: Find radius from equation

Given \((x - 3)^2+(y + 5)^2=25\), since \(r^2 = 25\), then \(r=\sqrt{25}=5\).

Step4: Find center from equation

For \((x + 1)^2+(y - 4)^2=36\), comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(h=-1\) and \(k = 4\), so the center is \((-1,4)\).

Step5: Recall circle - equation name

The formula \((x - h)^2+(y - k)^2=r^2\) is the standard - form equation of a circle.

Answer:

1. radius
2. \(h\); \(k\)
3. \(5\)
4. \((-1,4)\)
5. standard - form equation