Question

which point lies on the circle represented by the equation (x - 1)^2+(y + 2)^2 = 25?
a. (0,0)
b. (6,-2)
c. (3,1)
d. (0,-7)
what is the distance between the points (2,3) and (5,7)?
a. 5
b. 3
c. 6
d. 7
what is the equation of a circle with a center at the origin and radius of 5?
a. (x - 5)^2+(y + 5)^2 = 25
b. x^2 + y^2 = 5
c. x^2 + y^2 = 10
d. x^2 + y^2 = 25

Explanation:

Step1: Check point for first circle equation

Substitute \(x = 0,y = 0\) into \((x - 1)^2+(y + 2)^2\): \((0 - 1)^2+(0+ 2)^2=1 + 4=5
eq25\).

Step2: Try second - point

Substitute \(x = 6,y=-2\) into \((x - 1)^2+(y + 2)^2\): \((6 - 1)^2+(-2 + 2)^2=25+0 = 25\). So \((6,-2)\) lies on the circle.

Step3: Distance formula for second question

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 2,y_1=3,x_2 = 5,y_2 = 7\), so \(d=\sqrt{(5 - 2)^2+(7 - 3)^2}=\sqrt{9 + 16}=\sqrt{25}=5\).

Step4: Equation of circle for third question

The standard form of a circle with center \((h,k)\) and radius \(r\) is \((x - h)^2+(y - k)^2=r^2\). For a circle with center at the origin \((0,0)\) and \(r = 5\), the equation is \((x-0)^2+(y - 0)^2=5^2\), i.e., \(x^2+y^2=25\).

Answer:

1. B. \((6,-2)\)
2. A. 5
3. D. \(x^2 + y^2=25\)