Question

the point equidistant from all points on a circle is called the

the distance from the center of a circle to any point on the circle is the

the formula (x - h)^2+(y - k)^2 = r^2 represents the equation of a

in the equation (x - 3)^2+(y + 4)^2 = 25, the value of h is

in the equation (x - 3)^2+(y + 4)^2 = 25, the value of k is

the standard form of a circles equation includes the and radius.

the center of a circle with the equation (x + 1)^2+(y - 2)^2 = 16 is

the radius of a circle with the equation (x - 2)^2+(y + 3)^2 = 49 is

Explanation:

Step1: Recall circle - center definition

The point equidistant from all points on a circle is the center.

Step2: Recall radius definition

The distance from the center of a circle to any point on the circle is the radius.

Step3: Identify circle - equation form

The formula \((x - h)^2+(y - k)^2=r^2\) represents the equation of a circle, where \((h,k)\) is the center and \(r\) is the radius.

Step4: Find \(h\) value in given equation

For the equation \((x - 3)^2+(y + 4)^2=25\), comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(h = 3\).

Step5: Find \(k\) value in given equation

For the equation \((x - 3)^2+(y + 4)^2=25\), comparing with \((x - h)^2+(y - k)^2=r^2\), we rewrite \(y+4\) as \(y-(-4)\), so \(k=-4\).

Step6: Recall circle - equation components

The standard - form of a circle's equation \((x - h)^2+(y - k)^2=r^2\) includes the center \((h,k)\) and radius \(r\).

Step7: Find center of given circle

For the equation \((x + 1)^2+(y - 2)^2=16\), comparing with \((x - h)^2+(y - k)^2=r^2\), we rewrite \(x + 1\) as \(x-(-1)\), so the center is \((-1,2)\).

Step8: Find radius of given circle

For the equation \((x - 2)^2+(y + 3)^2=49\), comparing with \((x - h)^2+(y - k)^2=r^2\), since \(r^2 = 49\), then \(r = 7\).

Answer:

1. center
2. radius
3. circle
4. 3
5. - 4
6. center
7. \((-1,2)\)
8. 7