Question

1. ( - 3, - 2 ) r θ x y 4. ( 5, 12 ) r θ x y

Explanation:

Assuming the problem is to find the radius \( r \) for the points on the circles (since the circles are centered at the origin, we can use the distance formula from the origin to the point \((x,y)\), which is \( r = \sqrt{x^2 + y^2} \)). Let's solve for both points:

For the point \((-3, -2)\) (first circle):

Step1: Identify \( x \) and \( y \)

Here, \( x = -3 \), \( y = -2 \).

Step2: Apply the distance formula from origin

The formula for the radius (distance from origin \((0,0)\) to \((x,y)\)) is \( r = \sqrt{x^2 + y^2} \). Substitute \( x = -3 \) and \( y = -2 \):
\( r = \sqrt{(-3)^2 + (-2)^2} \)

Step3: Calculate the squares

\( (-3)^2 = 9 \), \( (-2)^2 = 4 \). So, \( r = \sqrt{9 + 4} \)

Step4: Sum and take square root

\( 9 + 4 = 13 \), so \( r = \sqrt{13} \)

For the point \((5, 12)\) (second circle):

Step1: Identify \( x \) and \( y \)

Here, \( x = 5 \), \( y = 12 \).

Step2: Apply the distance formula from origin

The formula for the radius (distance from origin \((0,0)\) to \((x,y)\)) is \( r = \sqrt{x^2 + y^2} \). Substitute \( x = 5 \) and \( y = 12 \):
\( r = \sqrt{5^2 + 12^2} \)

Step3: Calculate the squares

\( 5^2 = 25 \), \( 12^2 = 144 \). So, \( r = \sqrt{25 + 144} \)

Step4: Sum and take square root

\( 25 + 144 = 169 \), and \( \sqrt{169} = 13 \), so \( r = 13 \)

For \((-3, -2)\):

Answer:

\( \sqrt{13} \)

For \((5, 12)\):