Question

A golf course features a driving range with a circular boundary represented by the equation (x−3)2+(y+2)2=81
. Golfers receive a score whenever their ball lands inside the boundary. Four balls are hit, landing at the coordinates listed below. Which ball lands inside the boundary?(1 point)
Responses

(−1,−11)
open paren negative 1 comma negative 11 close paren

(1,7)
open paren 1 comma 7 close paren

(−2,4)
open paren negative 2 comma 4 close paren

(11,5)

Answer 1

To determine which ball lands inside the boundary of the driving range, we need to analyze the given equation of the circular boundary:

\[
(x - 3)^2 + (y + 2)^2 = 81
\]

This is the equation of a circle with center at \((3, -2)\) and a radius of \(9\) (since \(\sqrt{81} = 9\)).

A point \((x, y)\) is inside the circle if the left side of the equation is less than \(81\):

\[
(x - 3)^2 + (y + 2)^2 < 81
\]

Now, let's evaluate each of the four points:

1. For \((-1, -11)\):
\[
(-1 - 3)^2 + (-11 + 2)^2 = (-4)^2 + (-9)^2 = 16 + 81 = 97
\]
Since \(97 > 81\), this point is outside the boundary.

2. For \((1, 7)\):
\[
(1 - 3)^2 + (7 + 2)^2 = (-2)^2 + (9)^2 = 4 + 81 = 85
\]
Since \(85 > 81\), this point is outside the boundary.

3. For \((-2, 4)\):
\[
(-2 - 3)^2 + (4 + 2)^2 = (-5)^2 + (6)^2 = 25 + 36 = 61
\]
Since \(61 < 81\), this point is inside the boundary.

4. For \((11, 5)\):
\[
(11 - 3)^2 + (5 + 2)^2 = (8)^2 + (7)^2 = 64 + 49 = 113
\]
Since \(113 > 81\), this point is outside the boundary.

The only ball that lands inside the boundary is at the coordinate **(-2, 4)**.