Find a point on the y -axis which is equidistant from the points A(6,5) and B(−4,3).
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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. So, 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(6,5) and B(−4,3). 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 into the formula
Substitute the coordinates of P(0, y), A(6,5), and B(−4,3) into the squared distance formula. For PA2, we use x1=0,y1=y and x2=6,y2=5. For PB2, we use x3=0,y3=y and x4=−4,y4=3.
Step 5: Simplify and expand the equation
First, calculate the squared terms for the x -coordinates: (−6)2=36 and (4)2=16. Then, expand the squared binomials using the formula (a−b)2=a2−2ab+b2: (y−5)2=y2−10y+25 and (y−3)2=y2−6y+9.
Step 6: Combine constant terms and variables
Combine the constant terms on both sides of the equation. On the left side, 36+25=61. On the right side, 16+9=25.
Step 7: Solve for y
Subtract y2 from both sides of the equation. Then, isolate the terms containing y on one side and the constant terms on the other side. Perform the arithmetic to find the value of y.
Step 8: State the coordinates of the point
Since we found y=9 and the point is on the y -axis, its coordinates are (0,9).