Question

find all points having an x - coordinate of 3 whose distance from the point (-1, -6) is 5.
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 3 be $(3,y)$, and the given point is $(-1,-6)$ with $d = 5$.

Step2: Substitute values into formula

Substitute $x_1=-1,y_1 = - 6,x_2=3,y_2=y$ and $d = 5$ into the distance formula:
\[5=\sqrt{(3-(-1))^2+(y - (-6))^2}\]
\[5=\sqrt{(4)^2+(y + 6)^2}\]
\[5=\sqrt{16+(y + 6)^2}\]

Step3: Square both sides

Squaring both sides to get rid of the square - root:
\[25=16+(y + 6)^2\]

Step4: Solve for $(y + 6)^2$

Subtract 16 from both sides:
\[(y + 6)^2=25 - 16=9\]

Step5: Solve for y

Take the square root of both sides:
\[y+6=\pm3\]
When $y + 6=3$, then $y=3 - 6=-3$.
When $y + 6=-3$, then $y=-3 - 6=-9$.

Answer:

$(3,-3),(3,-9)$