Question

point b has coordinates (5,1). the x - coordinate of point a is 0. the distance between point a and point b is 13 units. what are the possible coordinates of point a? the possible coordinates of point a are \boxed{}. (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 point \(A=(0,y)\) and point \(B=(5,1)\), and \(d = 13\). Substitute into the formula:
\[13=\sqrt{(5 - 0)^2+(1 - y)^2}\]

Step2: Square both sides

Square both sides of the equation to eliminate the square root:
\[13^2=(5)^2+(1 - y)^2\]
\[169 = 25+(1 - y)^2\]

Step3: Solve for \((1 - y)^2\)

Subtract 25 from both sides:
\[(1 - y)^2=169 - 25\]
\[(1 - y)^2 = 144\]

Step4: Take square roots

Take the square root of both sides:
\[1 - y=\pm12\]

Step5: Solve for \(y\)

Case 1: \(1 - y = 12\)
Subtract 1 from both sides: \(-y=12 - 1=11\), so \(y=- 11\)
Case 2: \(1 - y=-12\)
Subtract 1 from both sides: \(-y=-12 - 1=-13\), so \(y = 13\)

So the possible coordinates of point \(A\) are \((0,13)\) or \((0,-11)\)

Answer:

\((0,13)\) or \((0,-11)\)