Question

find all points on the x - axis that are 10 units from the point (3, - 5). all points on the x - axis that are 10 units from the point (3, - 5) are. (simplify your answer. type an ordered pair. use a comma to separate answers as needed.)

Explanation:

Step1: Set up the distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Points on the $x$-axis have the form $(x,0)$. Let $(x_1,y_1)=(x,0)$ and $(x_2,y_2)=(3,-5)$ and $d = 10$. Then $10=\sqrt{(x - 3)^2+(0+ 5)^2}$.

Step2: Square both sides

Squaring both sides of the equation $10=\sqrt{(x - 3)^2+25}$, we get $100=(x - 3)^2+25$.

Step3: Simplify the equation

Subtract 25 from both sides: $(x - 3)^2=100 - 25=75$.

Step4: Solve for x

Take the square - root of both sides: $x-3=\pm\sqrt{75}=\pm5\sqrt{3}$. Then $x = 3\pm5\sqrt{3}$.

Answer:

$(3 + 5\sqrt{3},0),(3 - 5\sqrt{3},0)$