Let's solve the problem step-by-step:
We need to find the points on the y-axis that are at a distance of 13 units from the point [tex]\((-5, 7)\)[/tex].
1. Understand the problem: We are looking for points [tex]\((0, y_2)\)[/tex] on the y-axis, and the distance between these points and the point [tex]\((-5, 7)\)[/tex] should be 13 units.
2. Distance formula: The distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
3. Apply the distance formula: Set the coordinates [tex]\((x_1, y_1) = (-5, 7)\)[/tex] and [tex]\((x_2, y_2) = (0, y_2)\)[/tex]. Set the distance [tex]\(d\)[/tex] as 13 units. Substitute these values into the distance formula:
[tex]\[ 13 = \sqrt{(0 - (-5))^2 + (y_2 - 7)^2} \][/tex]
[tex]\[ 13 = \sqrt{5^2 + (y_2 - 7)^2} \][/tex]
[tex]\[ 13 = \sqrt{25 + (y_2 - 7)^2} \][/tex]
4. Solve for [tex]\(y_2\)[/tex]:
[tex]\[ 13^2 = 25 + (y_2 - 7)^2 \][/tex]
[tex]\[ 169 = 25 + (y_2 - 7)^2 \][/tex]
[tex]\[ 169 - 25 = (y_2 - 7)^2 \][/tex]
[tex]\[ 144 = (y_2 - 7)^2 \][/tex]
5. Take the square root of both sides:
[tex]\[ \sqrt{144} = |y_2 - 7| \][/tex]
[tex]\[ 12 = |y_2 - 7| \][/tex]
This gives us two equations:
[tex]\[ y_2 - 7 = 12 \text{ or } y_2 - 7 = -12 \][/tex]
6. Solve each equation:
[tex]\[ y_2 - 7 = 12 \][/tex]
[tex]\[ y_2 = 19 \][/tex]
[tex]\[ y_2 - 7 = -12 \][/tex]
[tex]\[ y_2 = -5 \][/tex]
Therefore, the two points on the y-axis that are at a distance of 13 units from the point [tex]\((-5, 7)\)[/tex] are:
[tex]\[(0, 19)\][/tex] and [tex]\[(0, -5)\][/tex].
