Find a point on the y-axis that is at the same distance from the points (−5,2) and (9,−2).
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Step-by-Step Solution
Step 1: Define the point on the y-axis
A point on the y -axis always has its x -coordinate equal to 0. Therefore, we can represent the required point as P(0, y).
Step 2: Set up the distance condition
The problem states that the point P is equidistant from points A(−5,2) and B(9,−2). This means the distance from P to A is equal to the distance from P to B.
Step 3: Apply the distance formula
To simplify calculations, we square both sides of the distance equality, PA=PB, to remove the square roots. We then apply the distance formula, which states that the distance between two points (x1,y1) and (x2,y2) is (x2−x1)2+(y2−y1)2.
Step 4: Substitute the coordinates and simplify
Substitute the coordinates of P(0, y), A(−5,2), and B(9,−2) into the squared distance formula. Then, simplify the squared terms and expand the binomials using the formula (a±b)2=a2±2ab+b2.
Step 5: Solve for y
First, combine the constant terms on both sides. Then, subtract y2 from both sides. Collect the y terms on one side and the constant terms on the other side. Finally, divide to find the value of y.
Step 6: State the coordinates of the point
Since we found y=−7 and the point is on the y -axis, its coordinates are (0,−7).