Question

find all points having an x - coordinate of 4 whose distance from the point (-2,-1) is 10.
the point(s) is(are)
(type an ordered pair. use a comma to separate answers as needed.)

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Let the point with $x -$coordinate $4$ be $(4,y)$, and the given point be $(-2,-1)$ with $d = 10$.

Step2: Substitute values into formula

Substitute $x_1=-2,y_1 = - 1,x_2 = 4,y_2=y$ and $d = 10$ into the distance formula:
\[10=\sqrt{(4-(-2))^2+(y - (-1))^2}\]
\[10=\sqrt{(4 + 2)^2+(y + 1)^2}\]
\[10=\sqrt{36+(y + 1)^2}\]

Step3: Square both sides

Squaring both sides to get rid of the square - root:
\[100=36+(y + 1)^2\]

Step4: Isolate the squared term

Subtract 36 from both sides:
\[(y + 1)^2=100 - 36\]
\[(y + 1)^2=64\]

Step5: Solve for y

Take the square root of both sides:
\[y+1=\pm8\]
When $y + 1=8$, then $y=7$. When $y + 1=-8$, then $y=-9$.

Answer:

$(4,7),(4,-9)$