The point on x-axis which is equidistant from the points (5, -3) and (4, 2) is
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Step-by-Step Solution
Step 1: Define the point on the x-axis
A point on the x -axis always has its y -coordinate equal to 0. Let this point be P(x, 0). We are given two other points, A(5,−3) and B(4,2).
Step 2: Apply the equidistant condition
The problem states that point P is equidistant from points A and B. This means the distance from P to A is equal to the distance from P to B.
Step 3: Use the distance formula
We use the distance formula to express PA and PB. The distance formula between two points (x1,y1) and (x2,y2) is given by (x2−x1)2+(y2−y1)2.
Step 4: Set up the equation
Substitute the coordinates of P(x, 0), A(5,−3), and B(4,2) into the distance formula and equate PA2 and PB2 to eliminate the square root. This gives us (x−5)2+(0−(−3))2=(x−4)2+(0−2)2.
Step 5: Solve for x
Expand and simplify the equation. The x2 terms cancel out. Collect like terms to solve for x. This leads to x=7.
Step 6: State the point
Since we found x=7, the point on the x -axis equidistant from the given points is (7,0).