Question

1. what is the distance between the points (2,3) and (5,7)?

a. 7
b. 3
c. 5
d. 6

1. what is the radius of the circle with the equation (x + 3)^2+(y - 2)^2 = 36?

a. 12
b. 3
c. 6
d. 9

1. what is the radius of the circle with the equation (x + 5)^2+(y - 6)^2 = 49?

a. 7
b. 14
c. 49
d. 5

Explanation:

Step1: Recall distance - formula

The distance formula 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, \(x_1 = 2,y_1 = 3,x_2 = 5,y_2 = 7\).
\[d=\sqrt{(5 - 2)^2+(7 - 3)^2}\]

Step2: Calculate the squares

\((5 - 2)^2=3^2 = 9\) and \((7 - 3)^2=4^2 = 16\). Then \(d=\sqrt{9 + 16}\).

Step3: Find the sum and square - root

\(9+16 = 25\), so \(d=\sqrt{25}=5\).

Step4: Recall circle - equation formula

The standard form of the equation of a circle is \((x - a)^2+(y - b)^2=r^2\), where \((a,b)\) is the center of the circle and \(r\) is the radius.

For the equation \((x + 3)^2+(y - 2)^2=36\), we can rewrite it as \((x-(-3))^2+(y - 2)^2 = 6^2\), so the radius \(r = 6\).

For the equation \((x + 5)^2+(y - 6)^2=49\), we can rewrite it as \((x-(-5))^2+(y - 6)^2=7^2\), so the radius \(r = 7\).

Answer:

1. c. 5
2. c. 6
3. a. 7